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by 

The  estate  of 
Herbert  W,  Rand 
January  9,  19^4 


I 

I 


^ihCAJ^Cc 


GROWTH  AND  FORM 


CAMBRIDGE   UNIVERSITY   PRESS 

C.  F.  CLAY,  Manager 

iLontion:  fetter  lane,  e.g. 

lEJinfeurs!) :  loo  PRINCES  STREET 


ip.tfa  HorJi:    G.  P.  PUTNAM'S  SONS 

53oml)nii,  ffalctitta  ant)  ffflaliraB:    MACMILI.AN  AND  Co.,  Ltd. 

arovonto:    J.    M.    DENT   AND   SONS,    Ltd. 

Coiuio:    THK,   MARUZEN-KABUSHIKI-KAISHA 


All  rights  reserved 


LIST   OF   ILLUSTRATIONS 


XV 


FIG. 

381, 
383. 

384. 

385, 
387. 
388-92. 


2.     A  similar  comparison  of  Diodon  and  Orthagoriscus       .         .         . 

The   same   of   various   crocodiles:     C.   2^o^osus,   C.   americanus   and 
Notosuchus  terrestris     ......... 

The  pelvic  girdles  of  Stegosaurus  and  Camptosaurus 
6.     The  shoulder-girdles  of  Cryptocleidus  and  of  Ichthyosaurus  . 

The  skulls  of  Dimorphodon  and  of  Pteranodon  .... 


The  pelves  of  Archaeopteryx  and  of  Apatornis  compared,  and  a 
method  illustrated  whereby  intermediate  configurations  may  be 
found  by  interpolation  (G.  Heilmann)         .... 

393.  The  same  pelves,  together  with  three  of  the  intermediate  or  inter 

polated  forms       ......... 

394,  5.     Comparison  of  the  skulls  of  two  extinct  rhinoceroses,  Hyrachyii 

and  Aceratherium  (Osbom)  ...... 

396.     Occipital  views  of  various  extinct  rhinoceroses  (do.) 

397—400.     Comparison  with  each  other,  and  with  the  skull  of  Hyrachyus,  of 

the  skulls  of  Titanotherium,  tapir,  horse  and  rabbit 
401,  2.     Coordinate  diagrams  of  the  skulls  of  Eohippus  and  of  Equus,  with 

various  actual  and  hypothetical  intermediate  types  (Heilmann) 

403.  A  comparison  of  various  human  scapulae  (Dwight) 

404.  A  human  skull,  inscribed  in  Cartesian  coordinates 

405.  The  same  coordinates  on  a  new  projection,  adapted  to  the  skull  of 

the  chimpanzee    .         .         .         .         .         . 

406.  Chimpanzee's  skull,  inscribed  in  the  network  of  Fig.  405 

407.  8.     Corresponding  diagrams  of  a  baboon's  skull,  and  of  a  dog's 


PAGE 

751 

753 
754 
755 
756 


757-9 
760 

761 

762 

763,4 

765-7 
769 
770 

770 

771 

771,3 


"Cum  formarum  naturalium  et  corporalium  esse  non  consistat  iiisi  in 
unione  ad  materiam,  ejusdem  agentis  esse  videtur  eas  producere  cujus  est 
materiam  transmutare.  Secundo,  quia  cum  hujusmodi  formae  non  excedant 
virtutem  et  ordinem  et  facultatem  principiorum  agentium  in  natura,  nulla 
videtur  necessitas  eorum  originem  in  principia  reducere  altiora."  x'^quinas, 
De  Pat.  Q.  ui,  a,  11.     (Quoted  in  Brit.  Assoc.  Address,  Section  D,  1911.) 

"...I  would  that  all  other  natural  phenomena  might  similarly  be  deduced 
from  mechanical  principles.  For  many  things  move  me  to  suspect  that 
everythmg  depends  upon  certain  forces,  in  wtue  of  which  the  particles  of 
bodies,  through  forces  not  yet  understood,  are  either  impelled  together  so  as 
to  cohere  in  regular  figures,  or  are  repelled  and  recede  from  one  another." 
Newton,  in  Preface  to  the  Principia.  (Quoted  by  ]VIr  W.  Spottiswoode, 
Brit.  Assoc.  Presidential  Address,  1878.) 

"When  Science  shall  have  subjected  all  natmal  phenomena  to  the  laws  of 
Theoretical  Mechanics,  when  she  shall  be  able  to  predict  the  result  of  every 
combination  as  unerringly  as  Hamilton  predicted  conical  refraction,  or  Adams 
revealed  to  us  the  existence  of  Neptune, — that  we  cannot  say.  That  day 
may  never  come,  and  it  is  certainly  far  in  the  dim  futiu-e.  We  may  not 
anticipate  it,  we  may  not  even  call  it  possible.  But  none  the  less  are  we 
bound  to  look  to  that  day,  and  to  labour  for  it  as  the  crowning  triumph  of 
Science : — when  Theoretical  Mechanics  shall  be  recognised  as  the  key  to  every 
physical  enigma,  the  chart  for  every  traveller  through  the  dark  Infinite  of 
Nature."     J.  H.  Jellett,  in  Brit.  Assoc.  Address,  Section  A,  1874. 


ON 


GROWTH    AND    FORM 


BY 


D'ARCY    WENTWORTH    THOMPSON 


^ca^ 


l^l  LIBRARY  1^ 


MASS. 


^/ 


Cambridge : 

at  the  University   Press 

1917 


■'The  reasonings  about  the  wonderful  and  intricate  operations 
of  nature  are  so  full  of  uncertainty,  that,  as  the  Wise-man  truly 
observes,  hardly  do  we  guess  aright  at  the  things  that  are  upon 
earth,  and  ivith  labour  do  we  find  the  things  that  are  before  us." 
Stephen  Hales,  Vegetable  Staticks  (1727),  p.  318,  1738. 


PREFATORY   NOTE 

THIS  book  of  mine  has  little  need  of  preface,  for  indeed  it  is 
"  all  preface  "  from  beginning  to  end.  I  have  written  it  as 
an  easy  introduction  to  the  study  of  organic  Form,  by  methods 
which  are  the  common-places  of  physical  science,  which  are  by 
no  means  novel  in  their  application  to  natural  history,  but  which 
nevertheless  naturalists  are  httle  accustomed  to  employ. 

It  is  not  the  biologist  with  an  inkling  of  mathematics,  but 
the  skilled  and  learned  mathematician  who  must  ultimately 
deal  with  such  problems  as  are  merely  sketched  and  adumbrated 
here.  I  pretend  to  no  mathematical  skill,  but  I  have  made  what 
use  I  could  of  what  tools  I  had ;  I  have  dealt  with  simple  cases, 
and  the  mathematical  methods  which  I  have  introduced  are  of 
the  easiest  and  simplest  kind.  Elementary  as  they  are,  my  book 
has  not  been  written  without  the  help — the  indispensable  help — 
of  many  friends.  Like  Mr  Pope  translating  Homer,  when  I  felt 
myself  deficient  I  sought  assistance !  And  the  experience  which 
Johnson  attributed  to  Pope  has  been  mine  also,  that  men  of 
learning  did  not  refuse  to  help  me. 

My  debts  are  many,  and  I  will  not  try  to  proclaim  them  all : 
but  I  beg  to  record  my  particular  obligations  to  Professor  Claxton 
Fidler,  Sir  George  Greenhill,  Sir  Joseph  Larmor,  and  Professor 
A.  McKenzie ;  to  a  much  younger  but  very  helpful  friend, 
Mr  John  Marshall,  Scholar  of  Trinity;  lastly,  and  (if  I  may  say 
so)  most  of  all,  to  my  colleague  Professor  William  Peddie,  whose 
advice  has  made  many  useful  additions  to  my  book  and  whose 
criticism  has  spared  me  many  a  fault  and  blunder. 

I  am  under  obligations  also  to  the  authors  and  publishers  of 
many  books  from  which  illustrations  have  been  borrowed,  and 
especially  to  the  following: — 

To  the  Controller  of  H.M.  Stationery  Office,  for  leave  to 
reproduce  a  number  of  figures,  chiefly  of  Foraminifera  and  of 
Eadiolaria,  from  the  Reports  of  the  Challenger  Expedition. 


vi  PREFATORY  NOTE 

To  the  Council  of  the  Royal  Society  of  Edinburgh,  and  to  that 
of  the  Zoological  Society  of  London : — the  former  for  letting  me 
reprint  from  their  Transactions  the  greater  part  of  the  text  and 
illustrations  of  my  concluding  chapter,  the  latter  for  the  use  of  a 
number  of  figures  for  my  chapter  on  Horns. 

To  Professor  E.  B.  Wilson,  for  his  well-known  and  all  but 
indispensable  figures  of  the  cell  (figs.  42 — 51,  53) ;  to  M.  A.  Prenant, 
for  other  figures  (41,  48)  in  the  same  chapter;  to  Sir  Donald 
MacAhster  and  Mr  Edwin  Arnold  for  certain  figures  (335 — 7), 
and  to  Sir  Edward  Schafer  and  Messrs  Longmans  for  another  (334), 
illustrating  the  minute  trabecular  structure  of  bone.  To  Mr 
Gerhard  Heilmann,  of  Copenhagen,  for  his  beautiful  diagrams 
(figs.  388-93,  401,  402)  included  in  my  last  chapter.  To  Pro- 
fessor Claxton  Fidler  and  to  Messrs  Grifl&n,  for  letting  me  use, 
with  more  or  less  modification  or  simplification,  a  number  of 
illustrations  (figs.  339 — 346)  from  Professor  Fidler's  Textbook  of 
Bridge  Construction.  To  Messrs  Blackwood  and  Sons,  for  several 
cuts  (figs.  127 — 9,  131,  173)  from  Professor  Alleyne  Nicholson's 
Palaeontology ;  to  Mr  Heinemann,  for  certain  figures  (57, 122, 123, 
205)  from  Dr  Stephane  Leduc's  Mechanism  of  Life ;  to  Mr  A.  M. 
Worthington  and  to  Messrs  Longmans,  for  figures  (71,  75)  from 
A  Study  of  Splashes,  and  to  Mr  C.  R.  Darhng  and  to  Messrs  E. 
and  S.  Spon  for  those  (fig.  85)  from  Mr  Darhng's  Liquid  Drops 
and  Globules.  To  Messrs  Macmillan  and  Co.  for  two  figures 
(304,  305)  from  Zittel's  Palaeontology;  to  the  Oxford  University 
Press  for  a  diagram  (fig.  28)  from  Mr  J.  W.  Jenkinson's  Experi- 
mental Embryology;  and  to  the  Cambridge  University  Press  for 
a  number  of  figures  from  Professor  Henry  Woods's  Invertebrate 
Palaeontology ,  for  oife  (fig.  210)  from  Dr  Willey's  Zoological  Results, 
and  for  another  (fig.  321)  from  "  Thomson  and  Tait." 

Many  more,  and  by  much  the  greater  part  of  my  diagrams, 
I  owe  to  the  untiring  help  of  Dr  Doris  L.  Mackinnon,  D.Sc,  and 
of  Miss  Helen  Ogilvie,  M.A.,  B.Sc,  of  this  College. 

D'ARCY  WENTWORTH  THOMPSON. 

University  College,  Dundee. 
December,  1916. 


1 1    I  / !" 


CONTENTS 

CHAP.  PAGE 

I.  Introductory \ 

II.  On  Magnitude 16 

III.  The  Rate  of  Growth ,50 

IV.  On  the  Internal  Form  and  Structure  of  the  Cell      .  156 

V.  The  Forms  of  Cells             . 201 

VI.  A  Note  on  Adsorption 277 

VII.  The  Forms  of  Tissues,  or  Cell-aggregates     .        .        .  293 

VIII.  The  same  {continued) •  .         .  346 

IX.  On  Concretions,  Spicules,  and  Spicular  Skeletons       .  411 

X.  A  Parenthetic  Note  on  Geodetics            ....  488 

XI.  The  Logarithmic  Spiral 493 

XII.  The  Spiral  Shells  of  the  Foraminifera          .         .         .  587 

XIII.  The  Shapes  of  Horns,  and  of  Teeth  or  Tusks:    with 

A  Note  on  Torsion 612 

XIV.  On  Leaf-arrangement,  or  Phyllotaxis             .         .         .  635 

XV.  On  the  Shapes  of  Eggs,  and  of  certain  other  Hollow 

Structures 652 

XVI.  On  Form  and  Mechanical  Efficiency       ....  670 

XVII.  On  the  Theory  of  Transformations,  or  the  Comparison 

OF  Related  Forms 719 

Epilogue 778 

Index 780 


/o'?i'a 


LIST   OF   ILLUSTRATIONS 


FIG.  PAGE 

1.     Nerve-cells,  from  larger  and  smaller  animals  (Minot,   after  Irving 


Hardesty)     .......... 

2.  Relative  magnitudes  of  some  minute  organisms  (Zsigmondy)     . 

3.  Curves  of  growth  in  man  (Quetelet  and  Bowditch) 

4.  5.     Mean  annual  increments  of  stature  and  weight  in  man  (do.) 
6.     The  ratio,  throughout  life,  of  female  weight  to  male  (do.) 

7—9.     Curves  of  growi;h  of  child,  before  and  after  birth  (His  and  Riissow 

10.  Curve  of  growth  of  bamboo  (Ostwald,  after  Kraus) 

11.  Coefficients  of  variability  in  human  stature  (Boas  and  Wisslerl 

12.  Growth  in  weight  of  mouse  (Wolfgang  Ostwald) 

13.  /)•-).  cf  silkworm  fLuciani  and  Lo  Monaco) 
]i.     Do.  of  tadpole  (Ostwald,  after  Schapcr)    . 

15.  Larval  eels,  or  Leptocephali,  aiid  young  elver  (Joh. 

16.  Growth  in  length  of  Spirogyra  (Hofmeister) 

17.  Pulsations  of  growth  in  Crocus  (Bose) 

18.  Relative  growi;h  of  brain,  heart  and  body  of  man 

19.  Ratio  of  stature  to  span  of  arms  (do.) 

20.  Rates  of  growth  near  the  tip  of  a  bean-root  (Sach 

21.  22.     The  weight-length  ratio  of  the  plaice,  and  its 

changes        ....... 

23.  Variability  of  tail-forceps  in  earwigs  (Bateson) 

24.  Variability  of  body-length  in  plaice 

25.  Rate  of  growth  in  plants  in  relation  to  temperature  (Sachs) 

26.  Do.  in  maize,  observed  (Koppen),  and  calculated  curves     . 

27.  Do.  in  roots  of  peas  (Miss  I.  Leitch)         .... 

28.  29. 


Schmidt) 


Quetelet) 


37 

39 

61 

66,  69 

71 

74-6 

77 
80 
8S 
84 
85 
86 
87 


annual  periodic 


90 
94 
96 

99, 100 
104 
105 
109 
112 
113 


115,6 
119 
120 


Rate  of  growth  of  frog  in  relation  to  temperature  (Jenkinson 
after  O.  Hertwig),  and  calculated  curves  of  do. 

30.  Seasonal  fluctuation  of  rate  of  growth  in  man  (Daffner) 

31.  Do.  in  the  rate  of  growth  of  trees  (C.  E.  Hall) 

32.  Long-period   fluctuation   in   the   rate   of   growth   of   Arizona    trees 

(A.  E.  Douglass) 122 

33.  34.     The   varying   form   of   brine-shrimps   (Arte7nia),   in   relation   to 

salinity  (Abonyi)  .........  128,9 

35-39.     Curves  of  regenerative  growth  in  tadpoles'  tails  (M.  L.  Durbin)     140-145 

40.  Relation  between  amoimt  of  tail  removed,   amount  restored,   and 

time  required  foi  restoration  (M.  M.  Ellis)         ....       148 

41.  Caryokinesis  in  trout's  egg  (Prenant,  after  Prof.  P.  Bouin)  .  .  169 
42-51.  Diagrams  of  mitotic  cell-division  (Prof.  E.  B.  Wilson)  .  .  171-5 
52.     Chromosomes  in  course  of  splitting  and  separation  (Hatschek  and 

Flemmmg) 180 


LIST   OF  ILLUSTRATIONS 


IX 


of  a  spher 


FIG. 

53.     Annular  chromosomes  of  mole-cricket  (Wilson,  after  vom  Rath)     . 

54-56.     Diagrams  illustrating  a  hypothetic  field  of  force  in  caryokinesis 

(Prof.  W.  Peddie) 

57.  An  artificial  figure  of  caryokinesis  (Leduc) 

58.  A  segmented  egg  of  Cerebratidus  (Prenant,  after  Coe) 

59.  Diagram  of  a  field  of  force  with  two  like  poles 

60.  A  budding  yeast-cell  ...... 

61.  The  roulettes  of  the  conic  sections    .... 

62.  Mode  of  development  of  an  unduloid  from  a  cylindrical  tube 
63-65.     Cylindrical,  unduloid,  nodoid  and  catenoid  oil-globules  (Plateau) 

Diagram  of  the  nodoid,  or  elastic  curve 

Diagram  of  a  cylinder  capped  by  the  corresponding  portion 

A  liquid  cylinder  breaking  up  into  spheres 

The  same  phenomenon  in  a  protoplasmic  cell  of  Trianea 

Some  phases  of  a  splash  (A.  M.  Worthington) 

A  breaking  wave  (do.)        ...... 

The  calycles  of  some  campanularian  zoophytes 

A  flagellate  monad,  Disiigma  proteiis  (Saville  Kent) 

Nocliluca  miliaris,  diagrammatic         .... 

Various  species  of  Vorticella  (Saville  Kent  and  others) 

Various  species  of  Salpingoeca  (do.) 

Species  of  Tintinnus,  Dinobryon  and  Codonella  (do.) 

The  tube  or  cup  of  Vaginicola  ... 

The  same  of  Folliculina     ...... 

Trachelophyllum  (Wveszniowski)  .... 

Trichodina  j)e,diculus  .         .  ... 

Dinenympha  gracilis  (Leidy)       ..... 

A  "collar-cell"  of  Codosiga        ..... 

Various  species  of  Lagtna  (Brady)     . 

Hanging  drops,  to  illustrate  the  unduloid  form  (C.  R.  Darling) 

Diagram  of  a  fluted  cylinder     .... 

Nodosaria  scalar i.?  (Brady)  .... 

Fluted  and  pleated  gonangia  of  certain  Campanularians  (AUman) 
Various  species  of  Nodosaria,  Sagrina  and  Rheophax  (Brady) . 
Trypanosoma  tineae  and  Spirochaeta  anodonfae,  to  shew  undulatin 

membranes  (Minchin  and  Fantham)  .... 

Some  species  of  Tricliomastix  and  Trichowonas  (Kofoid) 
Herpetomonas  assuming  the  undulatory  membrane  of  a  Trypanosome 

(D.  L.  Mackinnon) 

Diagram  of  a  human  blood-corpuscle 

Sperm-cells  of  decapod  crustacea,  Inachus  and  Galathea  (Koltzoff) 

The  same,  in  saline  solutions  of  varying  density  (do.) 

A  sperm-cell  of  Dromia  (do.)      .....•• 

Chondriosomes  in  cells  of  kidney  and  pancreas  (Barratt  and  Mathews 

Adsorptive  concentration  of  potassium  salts  in  various  plant-cell 

(Macallum)  ......••• 

99-101.     Equilibrium  of  surface-tension  in  a  floating  drop 

102.  Plateau's  "bourrelet"  in  plant-cells;  diagrammatic  (Berthold) 

103.  Parenchyma  of  maize,  shewing  the  same  phenomenon     . 


66. 
67. 
68. 
69. 
70. 
71. 
72. 
73. 
74. 
75. 
76. 
77. 
78. 
79. 
80. 
81. 
82. 
83. 
84. 
85. 
86. 
87. 
88. 
89. 
90. 

91. 

92. 

93. 
94. 
95. 
96. 
97. 
98. 


PAGE 
181 

182-4 
186 
189 
189 
213 
218 
220 

222,  3 
224 
226 
227 
234 
235 
236 
237 
246 
246 
247 
248 
248 
248 
249 
249 
252 
253 
254 
256 
257 
260 
262 
262 
263 

266 
267 

268 
271 
273 
274 
275 
285 

290 

294,5 

298 

298 


LIST   OF   ILLUSTEATIONS 


FIG. 

104,  5.     Diagrams  of  the  partition -wall  between  two  soap-biibbles 

106.  Diagram  of  a  partition  in  a  conical  cell 

107.  Chains  of  cells  in  Nostoc,  Anabaena  and  other  low  algae 

108.  Diagram  of  a  symmetrically  divided  soap-bubble     ... 

109.  Arrangement  of  partitions  in  dividing  spores  of  Pellia  (Campbell) 

110.  Cells  of  Dictyota  (Reinke) 

111.  2.     Terminal  and  other  cells  of  Chara,  and  young  antheridium  of  do 

113.  Diagram  of  cell-walls  and   partitions   under  various  conditions  of 

tension  ......... 

114,  5.     The  partition-sui'faces  of  three  interconnected  bubbles 

116.  Diagram  of  four  interconnected  cells  or  bubbles 

117.  Varioiis  configurations  of  four  cells  in  a  frog's  egg  (Rauber) 

1 18.  Another  diagram  of  two  conjoined  soap-bubbles 

119.  A  froth  of  bubbles,  shewing  its  outer  or  "epidermal"  layer 

120.  A  tetrahedron,  or  tetrahedral  system,  shewing  its  centre  of  symmetry 

121.  A  group  of  hexagonal  cells  (Bonanni)        .... 

122.  3.     Artificial  cellular  tissues  (Leduc)  ..... 

124.  Epidermis  of  Oirardia  (Goebel)  ..... 

125.  Soap-fioth,  and  the  same  under  compression  (Rhumbler) 

126.  Epidermal  cells  of  Elodea  canadensis  (Berthnld) 

127.  Lithostrotion  Martini  (Nicholson)        ..... 

128.  Cyathopkylbtm  hexagomim  (Nicholson,  after  Zittel)    . 

129.  Arachnophyllum  pentagonum  (Nicholson)    .... 

130.  Hdiolites  (Woods) 

131.  Confluent  septa  in   Thamnastraea   and   Comoseris  (Nicholson,  after 

Zittel) ... 

132.  Geometrical  construction  of  a  bee's  cell    .... 

133.  Stellate  cells  in  the  pith  of  a  rush ;  diagrammatic 

134.  Diagram  of  soap-films  formed  in  a  cubical  wire  skeleton  (Plateau) 

135.  Polar  furrows  in  systems  of  four  soap-bubbles  (Robert) 
136-8.     Diagrams  illustrating  the  division  of  a  cube  by  partitions  of  minimal 

area      ........... 

139.  Cells  from  hairs  of  Sphacelaria  (Berthold)  .... 

140.  The  bisection  of  an  isosceles  triangle  by  minimal  partitions    .       \ 

141.  The  similar  partitioning  of  spheroidal  and  conical  cells  . 

142.  S-shaped  partitions  from  cells  of  algae  and  mosses  (Reinke  and  others) 

143.  Diagrammatic  explanation  of  the  S-shaped  partitions 

144.  Development  of  Erythrotrichia  (Berthold) 

145.  Periclinal,  anticlinal  and  radial  partitioning  of  a  quadrant 

146.  Construction  for  the  minimal  partitioning  of  a  quadrant 

147.  Another  diagram  of  anticlinal  and  perichnal  partitions    . 

148.  Mode    of     segmentation    of     an    artificially    flattened    frog's    egg 

(Roux)  ......... 

149.  The  bisection,  by  minimal  partitions,  of  a  prism  of  small  angle 

150.  Comparative  diagram  of  the  various  modes  of  bisection  of  a  prismatic 

sector  ......... 

151.  Diagram  of  the  further  growth  of  the  two  halves  of  a  quadrantal  cell 

152.  Diagram  of  the  origin  of  an  epidermic  layer  of  cells 

153.  A  discoidal  cell  dividing  into  octants         .... 


PAGE 

299, 300 
300 


300 
301 
302 
303 
303 

304 
307,8 
309 
311 
313 
314 
317 
319 
320 
321 
322 
322 
325 
325 
326 
326 

327 
330 
335 
337 
341 

347-50 
351 
353 
353 
355 
356 
359 
359 
361 
362 


363 
364 

365 
367 
370 
371 


LIST    OF   ILLUSTRATIONS  xi 

FIG.  PAGE 

154.  A  germinating  spore  of  Riccia  (after  Campbell),  to  shew  the  manner 

of  space- partitioning  in  the  cellular  tissue          ....  372 

155,  6.     Theoretical  arrangement  of  successive  partitions  in  a  discoidal  coll     373 

157.  Sections  of  a  moss-embryo  (Kienitz-Gerloff) 374 

158.  Various  possible  arrangements  of  partitions  in  groups  of  four  to  eight 

cells !         ....  375 

159.  Three  modes  of  partitioning  in  a  system  of  six  cells      .         .         .  376 

160.  1.  Segmenting  eggs  of  Trochus  (Robert),  and  of  Cynthia  (Conklin)  .  377 
162.  Section  of  the  apical  cone  of  Salvinia  (Pringsheim)  .  .  .  377 
163, 4.     Segmenting  eggs  of  Pyrosom.a  (Korotneff),  and  of  Echinus  (l)riesch)  377 

165.  Segmenting  egg  of  a  cephalopod  (Watase) 378 

166,  7.     Eggs  segmenting  under  pressure:   of  Echinus  and  Nereis  (Driesch), 

and  of  a  frog  (Roux)           . 378 

168.  Various  arrangements  of  a  group  of  eight  cells  on  the  surface  of  a  frog's 

egg  (Rauber) 381 

169.  Diagram  of  the  partitions  and  interfacial  contacts  in  a  system  of  eight 

cells 383 

170.  Various  modes  of  aggregation  of  eight  oil-drops  (Roux)            .         .  384 

171.  Forms,  or  species,  of  Asterolampra  (Greville) 386 

172.  Diagrammatic  section  of  an  alcyonarian  polype        ....  387 

173.  4.     Sections  of  Heterophyllia  (Nicholson  and  Martin  Duncan)    .         .  388,9 

175.  Diagrammatic  section  of  a  ctenophore  (Eucharis)     ....  391 

176,  7.  Diagrams  of  the  construction  of  a  Pluteus  larva  .  .  -  392, 3 
178,  9.     Diagrams  of  the  development  of  stomata,  in  Sedum  and  in  the 

hyacinth       ...........  394 

180.  Various  spores  and  pollen-grains  (Berthold  and  others)    .         .         .  396 

181.  Spore  of  Anthoceros  (Campbell)           . 397 

182.  4,  9.     Diagrammatic  modes  of  division  of  a  cell  under  certain  conditions 

of  asymmetry       ..........  400-5 

183.  Development  of  the  embryo  of  Sphagnum  (Campbell)      .         .         .  402 

185.  The  gemma  of  a  moss  [do.)       ........  403 

186.  The  antheridium  of  Riccia  {do.) 404 

187.  Section  of  growing  shoot  of  Selaginella,  diagrammatic      .         .         .  404 

188.  An  embryo  of  Jungermannia  (Kienitz-Gerloff)           ....  404 

190.  Develoj^ment  of  the  sporangium  of  Osimmda  (Bower)      .         .         ■  406 

191.  Embryos  of  Phascum  and  of  Adiantum  (Kienitz-GerlolT)           .         .  408 

192.  A  section  of  Girardia  (Goebel) 408 

193.  An  antheridium  of  Pteris  (Strasburger)     ....-■  409 

194.  Spicules  of  Siphonogorgia  and  Anthogorgia  (Studer)  .  .  .  413 
195-7.     Calcospherites,  deposited  in  white  of  egg  (Harting)       .         .         ■  421,2 

198.  Sections  of  the  shell  of  My  a  (Carpenter) 422 

199.  Concretions,  or  spicules,  artificially  deposited  in  cartilage  (Hartmg)  423 

200.  Further  illustrations  of  alcyonarian  spicules:  Eunicea  (Studer)  .  424 
201-3.     Associated,  aggregated  and  composite  calcospherites  (Harting)     .  425,6 

204.  Harting's  "conostats" "^27 

205.  Liesegang's  rings  (Leduc)            .....•••  428 

206.  Relay-crystals  of  common  salt  (Bowman) 429 

207.  Wheel -like  crystals  in  a  colloid  medium  (do.)            .                  •         •  429 

208.  A  concentrically  striated  calcospherite  or  spherocrystal  (Harting)   .  432 


Xll 


LIST   OF   ILLUSTRATIONS 


FIG.  PAGE 

209.  Otoliths  of  plaice,  shewing  "age-rings"  (Wallace)     ....  432 

210.  Spicules,  or  calcospherites,  of  Astrosclera  (Lister)      ....  436 

211.  2.     C-  and  S-shaped  spicules  of  sponges  and  holothurians  (Sollas  and 

Theel) 442 

An  amphidisc  of  Hyalonema 442 

Spicules  of  calcareous,  tetractinellid  and  hexactinellid  sponges,  and 

of  various  holothurians  (Haeckel,  Schultze,  Sollas  and  Theel)  445-452 
Diagram  of  a  solid   body  confined   by  surface-energy  to   a  liquid 

boundary-film 460 

Astrorhiza  limicola  and  arenaria  (Brady)            .....  464 

A  nuclear  ^'reticulmn  plasmatigue'^  (Carnoy)     .....  468 

A  spherical  radiolarian,  Aulonia  hexagona  (Haeckel)         '.         .         .  469 

Actinomma  arcadophorum  (do.)            .......  469 

Ethmosphaera  conosipJionia  (do.)          .......  470 

Portions  of  shells  of  Cenosphaera  favosa  and  vesparia  (do.)      .         .  470 
Aulasfrum  triceros  (do.)       .         .         .         .         .         .         .         ,         .471 

Part  of  the  skeleton  of  CaniiorhapJ'is  (do.) 472 

A  Nassellarian  skeleton,  Callimitra.  carolotac  (do.)     ....  472 

228,  9.     Portions  of  Dictyocha  stapedia  (do.) 474 

230.     Diagram  to  illustrate  the  conformation  of  Callimitra        .         .         .  476 

Skeletons  of  various  radiolarians  (Haeckel) 479 

Diagrammatic  structure  of  the  skeleton  of  Doratas2ns  (do.)     .         .  481 

4.     Phatnaspis  cristata  (Haeckel),  and  a  diagram  of  the  same  .         .  483 

Phractasjris  prototypus  (Haeckel)         .......  484 

Annular  and  spiral  thickenings  in  the  walls  of  plant-cells        .         .  488 

A  radiograph  of  the  shell  of  Nautilus  (Green  and  Gardiner)      .         .  494 

A  spiral  foraminifer,  Globigerina  (Brady)            .....  495 


213. 
214-7 

218. 

219. 
220. 
221. 
222. 
223. 
224. 
225. 
226. 
227. 


231. 
2,32. 
233. 
235. 
236. 
237. 
238. 


239-42.     Diagrams  to  illustrate  the  development  or  growth  of  a  logarithmic 

spiral 407-501 

243.  A  helicoid  and  a  scorpioid  cyme       .......       502 

244.  An  Archimedean  spiral 503 

245-7.     More  diagrams  of  the  development  of  a  logarithmic  spiral  .         .  505,  6 
248-57.     Various  diagrams  illustrating  the  mathematical  theory  of  gnomons  508-13 

258.  A  shell  of  Haliotis,  to  shew  how  each  increment  of  the  shell  constitutes 

a  gnomon  to  the  preexisting  structure       .....       514 

259,  60.     Spiral  foraminifera,  Pulvinulina  and  Cristellaria,  to  illustrate  the 

same  principle      ..........  514, 5 

261.  Another  diagram  of  a  logarithmic  spiral  .....       517 

262.  A  diagram  of  the  logarithmic  spiral  of  Nautilus  (Moseley)       .         .       519 

263.  4.     Opercula  of  Turbo  and  of  Ncrita  (Moseley)  ....  521,  2 

265.  A  section  of  the  shell  of  Melo  ethiopicus  .....       525 

266.  SheUs  of  Harpa  and   Dolium,   to   illustrate   generating   curves   and 

generating  spirals         .........       526 

267.  D'Orbigny's  Helicometer    .         .  ......       529 

268.  Section  of  a  nautiloid  shell,  to  shew  the  "protoconch"  .         .         .       531 
269-73.     Diagrams  of  logarithmic  spirals,  of  varioiis  angles       .         .         .  532-5 

274,  6,  7.     Constructions  for  determining  the  angle  of  a  logarithmic  spiral       537,8 

275.  An  ammonite,  to  shew  its  corrugated  surface -pattern      .         .         .       537 
278-80.     Illustrations  of  the  "angle  of  retardation"  ....  542-4 


LIST   OF  ILLUSTRATIONS  xiii 

FIG.  ^  PAGE 

281.  A  shell' of  Macroscaphitps,  to  shew  change  of  curvature  .  .  550 

282.  Construction  for  determining  the  length  of  the  coiled  spire  .  .  551 

283.  Section  of  the  shell  of  Triton  corrugatus  (Woodward)       .  .  .  554 

284.  Lamellaria  perspicua  and  Sigaretus  halioioides  (do.)  .  .  .  555 

285.  6.  Sections  of  the  shells  of  Terebra  maculata  and  Trochus  nilolicvs  550.  (iO 
287-9.     Diagrams  illustrating  the  lines  of  growth  on  a  lamellibranch  shell  563-5 

290.  CaprineUa  adversa  (Woodward)  .......       507 

291.  Section  of  the  shell  of  Productus  (Woods)         .....       567 

292.  The  "skeletal  loop"  of  Terehratula  (do.) 508 

293.  4.  The  spiral  arms  of  Spirifer  and  of  Atrypa  (do.)  ....  5(i9 
295-7.  Shells  of  Cleodora,  Hyalam  and  other  pteropods  (Boas)  .  .  570, 1 
298,  9.     Coordinate  diagrams  of  the  shell-outline  in  certain  pteropods     .  572,3 

300.  Development  of  the  shell  of  Hyalaea  tridentata  (Tesch)    .         .         .       573 

301.  Pteropod  shells,  of  Cleodora  and  Hyalaea,  viewed  from  the  side  (Boas)       575 

302.  3.     Diagrams  of  septa  in  a  conical  shell 570 

304.  A  section  of  Nautikis,  shewing  the  logarithmic  spirals  of  the  septa  to 

which  the  shell- spiral  is  the  evolute  .         .         .         .         .581 

305.  Cast  of  the  interior  of  the  shell  of  Nautilus,  to  shew  the  contours  of 

the  septa  at  their  junction  with  the  shell-wall  .         .         .       582 

306.  Ammonites  Soinerhyi,  to  shew  septal  outlines  (Zittel,  after  Stcinmann 

and  Doderlein) 584 

307.  Suture-line  of  Pinacoceras  (Zittel,  after  Hauer)  ....  584 

308.  Shells  of  Hastigerina,  to  shew  the  "mouth"  (Brady)        .         ■         .  588 

309.  Nummulina  antiquior  (V.  von  Moller)        ......  591 

310.  Cornuspira  foliacea  and  Operculina  complanata  (Brady)    .         .         .  594 

311.  Miliolina  pulchella  and  liiuiaeana  (Brady)  .....  596 

312.  3.     Cyclammiria  cancellata  (do.),  and  diagrammatic  figure  of  the  same  596,  7 

314.  Orhulina  universa  (Brady)  ........  508 

315.  Cristellaria  reniformis  (do.)  ........  600 

316.  Discorhina  hertheloti  (do.)    .........  603 

317.  Textularia  trochus  and  concava  (do.)  ......  604 

318.  Diagrammatic  figure  of  a  ram's  horns  (Sir  V.  Brooke)    .         .         .  615 

319.  Head  of  an  Arabian  wild  goat  (Solater) 616 

320.  Head  of  Ouis  Ammon,  shewing  St  Venant's  curves  .         .         .  621 

321.  St  Venant's  diagram  of  a  triangular  prism  under  torsion  (Thomson 

and  Tait) 623 

322.  Diagram  of  the  same  phenomenon  in  a  ram's  horn          .         .         .  623 

323.  Antlers  of  a  Swedish  elk  (Lonnberg) 629 

324.  Head  and  antlers  of  Cervus  duvauceli  (Lydekker)    ....  630 

325.  6.     Diagrams  of  spiral  phyllotaxis  (P.  G.  Tait)  ....  644, 5 

327.  Further  diagrams  of  phyllotaxis,  to  shew  how  various  spiral  appear- 

ances may  arise  out  of  one  and  the  same  angular  leaf-divergence       648 

328.  Diagrammatic  outlines  of  various  sea-urchins  .....       664 

329.  30.  Diagrams  of  the  angle  of  branching  in  blood-vessels  (Hess)  .  667,8 
331,  2.     Diagrams  illustrating  the  flexure  of  a  beam         ....  674,  8 

333.  An  example  of  the  mode  of  arrangement  of  bast-fibres  in  a  plant-stem 

(Schwendener)      ..........       680 

334.  Section  of  the  head  of  a  femur,  to  shew  its  trabecular  structure 

(Schafer,  after  Robinson)    .......       681 


xiv  LIST   OF   ILLUSTRATIONS 

FIG.  PAGE 

335      Comparative  diagrams  of  a  crane-head  and  the  liead  of  a  femur 

(Culmann  and  H.  Meyer)    .         .  ....       682 

336.  Diagram  of  stress-lines  in  the  human  foot  (Sir  D.  MacAlister,  after 

H.  Meyer) 684 

337.  Trabecular  structure  of  the  05  cakis  (do.)  ....  685 

338.  Diagram  of  shearing-stress  in  a  loaded  pillar   .....  686 

339.  Diagrams  of  tied  arch,  and  bowstring  girder  (Fidler)       .         .         .  693 

340.  1.     Diagrams  of  a  bridge:    shewing  proposed  span,  the  corresponding 

stress-diagram  and  reciprocal  plan  of  construction  (do.)    .         .       696 

342.  A  loaded  bracket  and  its  reciprocal  construction-diagram  (Culmann)  .       697 

343,  4.     A  cantilever  bridge,  with  its  reciprocal  diagrams  (Fidler)    .         .       698 

345.  A  two -armed  cantilever  of  the  Forth  Bridge  (do.)   ....       700 

346.  A  two-armed  cantilever  with  load  distributed  over  two  pier-heads, 

as  in  the  quadrupedal  skeleton  ......       700 

347-9.     Stress-diagrams,  or  diagrams  of  bending  moments,  in  the  backbones 

of  the  horse,  of  a  Dinosaur,  and  of  Titanotherium    .         .         .  701-4 

350.  The  skeleton  of  Stegosaurus        ...  707 

351.  Bending-moments   in    a    beam    with    fixed    ends,    to    illustrate   the 

mechanics  of  chevron-bones         .......       709 

352.  3.     Coordinate  diagrams  of  a  circle,   and  its  deformation   into   an 

eUipse 729 

354.  Comparison,  by  means  of  Cartesian  coordinates,  of  the  cannon-bones 

of  various  ruminant  animals       .......       729 

355,  6.     Logarithmic  coordinates,  and  the  circle  of  Fig.  352  inscribed  therein  729, 31 
357,  8.     Diagrams  of  oblique  and  radial  coordinates  .....       731 

359.  Lanceolate,  ovate  and  cordate  leaves,  compared  by  the  help  of  radial 

coordinates  ..........       732 

360.  A  leaf  of  Begonia  daedalea         ........       733 

361.  A  network  of  logarithmic  spiral  coordinates     .         .         .         .         .       735 

362.  3.     Feet  of  ox,  sheep  and  giraffe,  compared  by  means  of  Cartesian 

coordinates 738, 40 

364,  6.     "Proportional  diagrams"  of  human  physiognomy  (Albert  Diirer)  740,2 

365.  Median  and  lateral  toes  of  a  tapir,  compared  by  means  of  rectangular 

and  oblique  coordinates       ........       741 

367,  8.     A  comparison  of  the  copepods  Oithona  and  Sapphirina       .         .       742 

369.  The  carapaces  of  certain  crabs,  Geryon,  Corystes  and  others,  compared 

by  means  of  rectilinear  and  curvilinear  coordinates  .         .       744 

370.  A  comparison  of  certain  amphipods,  Harpinia,   Stegocephalus  and 

Hyperia .       746 

371      The  calycles  of  certain  carapanularian  zoophytes,  inscribed  in  corre- 
sponding Cartesian  networks 747 

372.  The  calycles  of  certain  species  of  Aglaophenia,  similarly  compared  by 

means  of  curvilinear  coordinates         ......       748 

373,  4.     The  fishes  Argyropelecus  and  Sternoptyx,  compared  by  means  of 

rectangular  and  oblique  coordinate  systems       .         .  .       748 

375,  6.     Scarus  and  Pomacanthus,  similarly  compared  by  means  of  rect- 
angular and  coaxial  systems        .......       749 

377-80.     A  comparison  of  the  fishes  Polyprion,  Pseudopriacanthus,  Scorpaena 

and  Antigonia      ..........       750 


CHAPTER   I 

INTRODUCTORY 

Of  the  chemistry  of  his  day  and  generation,  Kant  declared 
that  it  was  "a  science,  but  not  science," — "eine  Wissenschaft, 
aber  nicht  Wissenschaft";  for  that  the  criterion  of  physical 
science  lay  in  its  relation  to  mathematics.  And  a  hundred  years 
later  Du  Bois  Reymond,  profound  student  of  the  many  sciences 
on  which  physiology  is  based,  recalled  and  reiterated  the  old 
saying,  declaring  that  chemistry  would  only  reach  the  rank  of 
science,  in  the  high  and  strict  sense,  when  it  should  be  found 
possible  to  explain  chemical  reactions  in  the  light  of  their  causal 
relation  to  the  velocities,  tensions  and  conditions  of  equihbrium 
of  the  component  molecules ;  that,  in  short,  the  chemistry  of  the 
future  must  deal  with  molecular  mechanics,  by  the  methods  and 
in  the  strict  language  of  mathematics,  as  the  astronomy  of  Newton 
and  Laplace  dealt  with  the  stars  in  their  courses.  We  know  how 
great  a  step  has  been  made  towards  this  distant  and  once  hopeless 
goal,  as  Kant  defined  it,  since  van't  Hoff  laid  the  firm  foundations 
of  a  mathematical  chemistry,  and  earned  his  proud  epitaph, 
Physicam  chemiae  adiunxit*. 

We  need  not  wait  for  the  full  reahsation  of  Kant's  desire,  in 
order  to  apply  to  the  natural  sciences  the  principle  which  he 
urged.  Though  chemistry  fall  short  of  its  ultimate  goal  in  mathe- 
matical mechanics,  nevertheless  physiology  is  vastly  strengthened 
and  enlarged  by  making  use  of  the  chemistry,  as  of  the  physics, 
of  the  age.  Little  by  little  it  draws  nearer  to  our  conception  of 
a  true  science,   with   each  branch  of  physical  science  which  it 

*  These  sayings  of  Kant  and  of  Du  Bois,  and  others  like  to  them,  have  been 
the  text  of  many  discourses :  see,  for  instance,  Stallo's  Concepts,  p.  21,  1882 ;  Hober, 
Biol  Cetdmlbl.  xix,  p.  284,  1890,  etc.     Cf.  also  Jellett,  Rep.  Brit.  Ass.  1874,  p.  1. 

T.  n.  1 


2  INTRODUCTORY  [ch. 

brings  into  relation  wdth  itself:  with  every  physical  law  and  every 
mathematical  theorem  which  it  learns  to  take  into  its  employ. 
Between  the  physiology  of  Haller,  fine  as  it  was,  and  that  of 
Helmholtz,  Ludwig,  Claude  Bernard,  there  was  all  the  difference 
in  the  world. 

As  soon  as  we  adventure  on  the  paths  of  the  physicist,  we 
learn  to  weigh  and  to  measure,  to  deal  with  time  and  space  and 
mass  and  their  related  concepts,  and  to  find  more  and  more 
our  knowledge  expressed  and  our  needs  satisfied  through  the 
concept  of  number,  as  in  the  dreams  and  visions  of  Plato  and 
Pythagoras;  for  modern  chemistry  would  have  gladdened  the 
hearts  of  those  great  philosophic  dreamers. 

But  the  zoologist  or  morphologist  has  been  slow,  where  the 
physiologist  has  long  been  eager,  to  invoke  the  aid  of  the  physical 
or  mathematical  sciences ;  and  the  reasons  for  this  difference  he 
deep,  and  in  part  are  rooted  in  old  traditions.  The  zoologist  has 
scarce  begun  to  dream  of  defining,  in  mathematical  language,  even 
the  simpler  organic  forms.  When  he  finds  a  simple  geometrical 
construction,  for  instance  in  the  honey-comb,  he  would  fain  refer 
it  to  psychical  instinct  or  design  rather  than  to  the  operation  of 
physical  forces ;  when  he  sees  in  snail,  or  nautilus,  or  tiny 
foraminiferal  or  radiolarian  shell,  a  close  approach  to  the  perfect 
sphere  or  spiral,  he  is  prone,  of  old  habit,  to  beheve  that  it  is 
after  all  something  more  than  a  spiral  or  a  sphere,  and  that  in 
this  "something  more"'  there  lies  what  neither  physics  nor 
mathematics  can  explain.  In  short  he  is  deeply  reluctant  to 
compare  the  living  with  the  dead,  or  to  explain  by  geometry  or 
by  dynamics  the  things  which  have  their  part  in  the  mystery  of 
life.  Moreover  he  is  little  inclined  to  feel  the  need  of  ^uch 
explanations  or  of  such  extension  of  his  field  of  thought.  He  is 
not  without  some  justification  if  he  feels  that  in  admiration  of 
nature's  handiwork  he  has  an  horizon  open  before  his  eyes  as 
wide  as  any  man  requires.  He  has  the  help  of  many  fascinating 
theories  within  the  bounds  of  his  own  science,  whieh,  though 
a  little  lacking  in  precision,  serve  the  purpose  of  ordering  his 
thoughts  and  of  suggesting  new  objects  of  enquiry.  His  art  of 
classification  becomes  a  ceaseless  and  an  endless  search  after  the 
blood-relationships  of  things  living,  and  the  pedigrees  of  things 


1]  THE   FINAL   CAUSE  3 

dead  and  gone.  The  facts  of  embryology  become  for  him,  as 
Wolff,  von  Baer  and  Fritz  Miiller  proclaimed,  a  record  not  only 
of  the  life-history  of  the  individual  but  of  the  annals  of  its  race. 
The  facts  of  geographical  distribution  or  even  of  the  migration  of 
birds  lead  on  and  on  to  speculations  regarding  lost  continents, 
sunken  islands,  or  bridges  across  ancient  seas.  Every  nesting 
bird,  every  ant-hill  or  spider's  web  displays  its  psychological 
problems  of  instinct  or  intelhgence.  Above  all,  in  things  both 
great  and  small,  the  naturalist  is  rightfully  impressed,  and  finally 
engrossed,  by  the  peculiar  beauty  which  is  manifested  in  apparent 
fitness  or  "adaptation," — the  flower  for  the  bee,  the  berry  for  the 
bird. 

Time  out  of  mind,  it  has  been  by  way  of  the  "final  cause," 
by  the  teleological  concept  of  "end,"  of  "purpose,"  or  of  "design," 
in  one  or  another  of  its  many  forms  (for  its  moods  are  many), 
that  men  have  been  chiefly  wont  to  explain  the  phenomena  of 
the  living  world;  and  it  will  be  so  while  men  have  eyes  to  see 
and  ears  to  hear  withal.  With  Galen,  as  with  Aristotle,  it  was 
the  physician's  way ;  with  John  Ray,  as  with  Aristotle,  it  was  the 
naturalist's  way ;  with  Kant,  as  with  Aristotle,  it  was  the  philo- 
sopher's way.  It  was  the  old  Hebrew  way,  and  has  its  splendid 
setting  in  the  story  that  God  made  "every  plant  of  the  field  before 
it  was  in  the  earth,  and  every  herb  of  the  field  before  it  grew." 
It  is  a  common  way,  and  a  great  way;  for  it  brings  with  it  a 
glimpse  of  a  great  vision,  and  it  lies  deep  as  the  love  of  nature 
in  the  hearts  of  men. 

Half  overshadowing  the  "efficient"  or  physical  cause,  the 
argument  of  the  final  cause  appears  in  eighteenth  century  physics, 
in  the  hands  of  such  men  as  Euler*  and  Maupertuis,  to  whom 
Leibniz  t  had  passed  it  on.  Half  overshadowed  by  the  me- 
chanical concept,  it  runs  through  Claude  Bernard's  LsQons  siir  les 

*  "  Quum  enim  mundi  universi  fabrica  sit  perfectissima,  atque  a  Creatore 
sapientissimo  absoluta,  nihil  omnino  in  mundo  contingit  in  quo  non  maximi 
minimive  ratio  quaepiam  eluceat;  quamobrem  dubium  prorsus  est  nullum  quin 
onines  mundi  effectus  ex  causis  finalibus,  ope  methodi  maximorum  et  minimorum, 
aeque  feliciter  determinari  queant  atque  ex  ipsis  causis  efficientibus."  Methodtis 
inveniendi,  etc.  1744  (cit.  Mach,  Science  of  Mechanics,  1902,  p.  455). 

t  Cf.  0pp.  (ed.  Erdmann),  p.  106,  "Bien  loin  d'exclure  les  causes  finales..., 
c'est  de  la  qu'il  faut  tout  deduire  en  Physique." 

1—2 


4  INTRODUCTORY  [ch. 

phenomenes  de  la  Vie*,  and  abides  in  much  of  modern  physio- 
logy f .  Inherited  from  Hegel,  it  dominated  Oken's  Natur'philosophie 
and  lingered  among  his  later  disciples,  who  were  wont  to  liken 
the  course  of  organic  evolution  not  to  the  straggling  branches  of 
a  tree,  but  to  the  building  of  a  temple,  divinely  planned,  and  the 
crowning  of  it  with  its  polished  minarets  J. 

It  is  retained,  somewhat  crudely,  in  modern  embryology,  by 
those  who  see  in  the  early  processes  of  growth  a  significance 
"rather  prospective  than  retrospective,"  such  that  the  embryonic 
phenomena  must  be  "referred  directly  to  their  usefulness  in 
building  the  body  of  the  future  animal§"  : — which  is  no  more,  and 
no  less,  than  to  say,  with  Aristotle,  that  the  organism  is  the  reXo^, 
or  final  cause,  of  its  own  processes  of  generation  and  development. 
It  is  writ  large  in  that  Entelechy||  which  Driesch  rediscovered, 
and  which  he  made  known  to  many  who  had  neither  learned  of  it 
from  Aristotle,  nor  studied  it  with  Leibniz,  nor  laughed  at  it  with 
Voltaire.  And,  though  it  is  in  a  very  curious  way,  we  are  told  that 
teleology  was  "refounded,  reformed  or  rehabilitated^"  by  Darwin's 
theory  of  natural  selection,  whereby  "every  variety  of  form  and 
colour  was  urgently  and  absolutely  called  upon  to  produce  its 
title  to  existence  either  as  an  active  useful  agent,  or  as  a  survival  ' 
of  such  active  usefulness  in  the  past.  But  in  this  last,  and  very 
important  case,  we  have  reached  a  "teleology"  without  a  reXo'i., 

*  Cf.  p.  162.  "La  force  vitale  dirige  des  phenomenes  qu'elle  ne  produit  pAs: 
les  agents  physiques  produisent  des  phenomenes  qu'ils  ne  dirigent  pas." 

t  It  is  now  and  then  conceded  with  reluctance.  Thus  Enriques,  a  learned 
and  philosophic  naturalist,  writing  "deUa  economia  di  sostanza  nelle  osse  cave" 
{Arch.f.  Entw.  Mech.  xx,  1906),  says  "una  certa  impronta  di  teleologismo  qua  e  la 
e  rimasta,  mio  malgrado,  in  questo  scritto." 

%  Cf.  Cleland,  On  Terminal  Forms  of  Life,  J.  Anat.  and  Phys.  xvni,  1884. 

§  Conklin,  Embryology  of  Crepidula,  Journ.  of  Marphol.  xm,  p.  203,  1897; 
Lillie,  F.  R.,  Adaptation  in  Cleavage,  Woods  Holl  Biol.  Lectures,  pp.  43-67,  1899. 

II  I  am  inclined  to  trace  back  Driesch' s  teaching  of  Entelechy  to  no  less  a 
person  than  Melanchthon.  When  Bacon  {de  Augm.  iv,  3)  states  with  disapproval 
that  the  soul  "has  been  regarded  rather  as  a  function  than  as  a  substance,"  R.  L. 
ElUs  points  out  that  ho  is  referring  to  Melanchthon' s  exposition  of  the  Aristotelian 
doctrine.  For  Melanchthon,  whose  view  of  the  peripatetic  philosophy  had  long 
great  influence  in  the  Protestant  Universities,  affirmed  that,  according  to  the  true 
view  of  Aristotle's  opinion,  the  soul  is  not  a  substance,  but  an  ivTeXexe^o.,  or 
function.  He  defined  it  as  dvva/jLis  qunedam  ciens  actiones — a  description  all  but 
identical  with  that  of  Claude  Bernard's  ""force  vitale.''' 

•1  Ray  Lankester,  Encijd.  Brit.  (9th  ed.),  art.  "Zoology,"  p.  806,  1889. 


I]  OF   TELEOLOGY   AND   MECHANISM  5 

as  men  like  Butler  and  Janet  have  been  pronipt  to  shew :  a  teleology 
in  which  the  final  cause  becomes  little  more,  if  anything,' than  the 
mere  expression  or  resultant  of  a  process  of  sifting  out  of  the 
good  from  the  bad,  or  of  the  better  from  the  worse,  in  short  of 
a  iprocess  of  mechanism*.  The  apparent  manifestations  of  "pur- 
pose" or  adaptation  become  part  of  a  mechanical  philosophy, 
according  to  which  "chaque  chose  finit  toujours  par  s'accommoder 
a  son  miheuf."  In  short,  by  a  road  which  resembles  but  is  not 
the  same  as  Maupertuis's  road,  we  find  our  way  to  the  very  world 
in  which  we  are  living,  and  find  that  if  it  be  not,  it  is  ever  tending 
to  become,  "the  best  of  all  possible  worlds {." 

But  the  use  of  the  teleological  principle  is  but  one  way,  not 
the  whole  or  the  only  way,  by  which  we  may  seek  to  learn  how 
things  came  to  be,  and  to  take  their  places  in  the  harmonious  com- 
plexity of  the  world.  To  seek  not  for  ends  but  for  "antecedents" 
is  the  way  of  the  physicist,  who  finds  "causes"  in  what  he  has 
learned  to  recognise  as  fundamental  properties,  or  inseparable 
concomitants,  or  unchanging  laws,  of  matter  and  of  energy.  In 
Aristotle's  parable,  the  house  is  there  that  men  may  live  in  it; 
but  it  is  also  there  because  the  builders  have  laid  one  stone  upon 
another:  and  it  is  as  a  mechanism,  or  a  mechanical  construction, 
that  the  physicist  looks  upon  the  world.  Like  warp  and  woof, 
mechanism  and  teleology  are  interwoven  together,  and  we  must 
not  cleave  to  the  one  and  despise  the  other;  for  their  union  is 
"rooted  in  the  very  nature  of  totality §." 

Nevertheless,  when  philosophy  bids  us  hearken  and  obey  the 
lessons  both  of  mechanical  and  of  teleological  interpretation,  the 
precept  is  hard  to  follow :  so  that  oftentimes  it  has  come  to  pass, 
just  as  in  Bacon's  day,  that  a  leaning  to  the  side  of  the  final 
cause  "hath  intercepted  the  severe  and  diligent  inquiry  of  all 

*  Alfred  Russel  Wallace,  especially  in  his  later  years,  relied  upon  a  direct  but 
somewhat  crude  teleology.  Cf.  his  World  of  Life,  a  Manifestation  of  Creative  Power, 
Directive  Mind  and  Ultimate  Purpose,  1910. 

t  Janet,  Les  Causes  Finales,  1876,  p.  350. 

t  The  phrase  is  Leibniz's,  in  his  Theodicee. 

§  Cf.  (int.  al.)  Bosanquet,  The  Meaning  of  Teleology,  Proc.  Brit.  Acad. 
1905-6,  pp.  235-245.  Cf.  also  Leibniz  (Discours  de  Metaphysiqtte;  Lettres  inedites, 
€d.  de  Careil,  1857,  p.  354;  cit.  Janet,  p.  643),  "L'un  et  Tautre  est  bon,  I'un  et 
I'autre  pent  etre  utile... et  les  auteurs  qui  suivent  ces  routes  differentes  ne  devraient 
point  se  maltraiter:    et  seq." 


6  INTRODUCTORY  [ch. 

real  and  physical  causes,"  and  has  brought  it  about  that  "the 
search  of  the  physical  cause  hath  been  neglected  and  passed  in 
silence."  So  long  and  so  far  as  "fortuitous  variation*"  and  the 
"survival  of  the  fittest"  remain  engrained  as  fundamental  and 
satisfactory  hypotheses  in  the  philosophy  of  biology,  so  long  will 
these  "satisfactory  and  specious  causes"  tend  to  stay  "severe  and 
diligent  inquiry,"  "to  the  great  arrest  and  prejudice  of  future 
discovery." 

The  difficulties  which  surround  the  concept  of  active  or  "  real " 
causation,  in  Bacon's  sense  of  the  word,  difficulties  of  which 
Hume  and  Locke  and  Aristotle  were  little  aware,  need 'scarcely 
hinder  us  in  our  physical  enquiry.  As  students  of  mathematical 
and  of  empirical  physics,  we  are  content  to  deal  with  those  ante- 
cedents, or  concomitants,  of  our  phenomena,  without  which  the 
phenomenon  does  not  occur, — with  causes,  in  short,  which,  aliae 
ex  aliis  aftae  et  necessitate  nexae,  are  no  more,  and  no  less,  than 
conditions  sine  qua  non.  Our  purpose  is  still  adequately  fulfilled : 
inasmuch  as  we  are  still  enabled  to  correlate,  and  to  equate,  our 
particular  phenomena  with  more  and  ever  more  of  the  physical 
phenomena  around,  and  so  to  weave  a  web  of  connection  and 
interdependence  which  shall  serve  our  turn,  though  the  meta- 
physician withhold  from  that  interdependence  the  title  of  causality. 
We  come  in  touch  with  what  the  schoolmen  called  a  ratio 
cognoscendi,  though  the  true  ratio  efficiendi  is  still  enwrapped  in 
many  mysteries.  And  so  handled,  the  quest  of  physical  causes 
merges  with  another  great  Aristotelian  theme, — the  search  for 
relations  between  things  apparently  disconnected,  and  for  "  simili- 
tude in  things  to  common  view  unlike."  Newton  did  not  shew 
the  cause  of  the  apple  falling,  but  he  shewed  a  similitude  between 
the  apple  and  the  stars. 

Moreover,  the  naturahst  and  the  physicist  will  continue  to 
speak  of  "causes,"  just  as  of  old,  though  it  may  be  with  some 
mental  reservations :  for,  as  a  French  philosopher  said,  in  a 
kindred    difficulty:     "ce   sont    la    des    manieres    de    s'exprimer, 

*  The  reader  will  understand  that  I  speak,  not  of  the  "severe  and  diligent 
inquiry"  of  variation  or  of  "fortuity,"  but  merely  of  the  easy  assumption  that 
these  phenomena  are  a  sufficient  basis  on  which  to  rest,  with  the  all-powerful 
help  of  natural  selection,  a  theory  of  definite  and  progressive  evolution. 


I]  OF   TELEOLOGY   AND   MECHANISM  7 

et    si    elles    sont    interdites    il    faut   renoncer    a    parler    de    ces 
choses." 

The  search  for  differences  or  essential  contrasts  between  the 
phenomena  of  organic  and  inorganic,  of  animate  and  inanimate 
things  has  occupied  many  mens'  minds,  while  the  search  for 
community  of  principles,  or  essential  simiUtudes,  has  been  followed 
by  few;  and  the  contrasts  are  apt  to  loom  too  large,  great  as 
they  may  be.  M.  Dunan,  discussing ^the  "Probleme  de  la  Vie*" 
in  an  essay  which  M.  Bergson  greatly  commends,  declares:  "Les 
lois  physico-chimiques  sont  aveugles  et  brutales ;  la  oil  elles 
regnent  seules,  au  lieu  d'un  ordre  et  d'un  concert,  il  ne  pent  y 
avoir  qu'incoherence  et  chaos."  But  the  physicist  proclaims 
aloud  that  the  physical  phenomena  which  meet  us  by  the  way 
have  their  manifestations  of  form,  not  less  beautiful  and  scarce 
less  varied  than  those  which  move  us  to  admiration  among  living 
things.  The  waves  of  the  sea,  the  Kttle  ripples  on  the  shore,  the 
sweeping  curve  of  the  sandy  bay  between  its  headlands,  the 
outUne  of  the  hills,  the  shape  of  the  clouds,  all  these  are  so  many 
riddles  of  form,  so  many  problems  of  morphology,  and  all  of 
them  the  physicist  can  more  or  less  easily  read  and  adequately 
solve :  solving  them  by  reference  to  their  antecedent  phenomena, 
in  the  material  system  of  mechanical  forces  to  which  they  belong, 
and  to  which  we  interpret  them  as  being  due.  They  have  also, 
doubtless,  their  immanent  teleological  significance;  but  it  is  on 
another  plane  of  thought  from  the  physicist's  that  we  contemplate 
their  intrinsic  harmony  and  perfection,  and  "see  that  they  are 
good." 

Nor  is  it  otherwise  with  the  material  forms  of  hving  things. 
Cell  and  tissue,  shell  and  bone,  leaf  and  flower,  are  so  many 
portions  of  matter,  and  it  is  in  obedience  to  the  laws  of  physics 
that  their  particles  have  been  moved,  moulded  and  conformed  f. 

*  Revue  Philosophique.  xxxiii,  1892. 

t  This  general  prineiple  was  clearly  grasped  by  Dr  George  Rainey  (a  learned 
physician  of  St  Bartholomew's)   many  years  ago,   and  expressed  in  such  words 

as  the  following  :   " it  is  illogical  to  suppose  that  in  the  case  of  vital  organisms 

a  distinct  force  exists  to  produce  results  perfectly  withm  the  reach  of  physical 
agencies,  especially  as  hi  many  instances  no  end  could  be  attained  were  that  the 
case,  but  that  of  opposing  one  force  by  another  capable  of  effecting  exactly 
the  same  purpose."  (On  Artificial  Calculi,  Q.J. M.S.  (Trans.  Microsc.  Soc),  vi, 
p.  49,    1858.)     Cf.  also  Helmholtz.  Infm  rit..  p.  9. 


8  INTRODUCTORY  [ch. 

They  are  no  exception  to  the  rule  that  0eo<?  aei  yew/jLerpel.  Their 
problems  of  form  are  in  the  first  instance  mathematical  problems, 
and  their  problems  of  growth  are  essentially  physical  problems; 
and  the  morphologist  is,  ipso  facto,  a  student  of  physical  science. 

Apart  from  the  physico-chemical  problems  of  modern  physio- 
logy, the  road  of  physico-mathematical  or  dynamical  investigation 
in  morphology  has  had  few  to  follow  it ;  but  the  pathway  is  old. 
The  way  of  the  old  Ionian  physicians,  of  Anaxagoras*,  of 
Empedocles  and  his  disciples  in  the  days  before  Aristotle,  lay 
just  by  that  highwayside.  It  was  Gahleo's  and  BorelU's  way. 
It  was  little  trodden  for  long  afterwards,  but  once  in  a  while 
Swammerdam  and  Reaumur  looked  that  way.  And  of  latCT 
years,  Moseley  and  Meyer,  Berthold,  Errera  and  Roux  have 
been  among  the  little  band  of  travellers.  We  need  not  wonder 
if  the  way  be  hard  to  follow,  and  if  these  wayfarers  have  yet 
gathered  httle.  A  harvest  has  been  reaped  by  others,  and  the 
gleaning  of  the  grapes  is  slow. 

It  behoves  us  always  to  remember  that  in  physics  it  has  taken 
great  men  to  discover  simple  things.  They  are  very  great  names 
indeed  that  we  couple  with  the  explanation  of  the  path  of  a  stone, 
the  droop  of  a  chain,  the  tints  of  a  bubble,  the  shadows  in  a  cup. 
It  is  but  the  slightest  adumbration  of  a  dynamical  morphology 
that  we  can  hope  to  have,  until  the  physicist  and  the  mathematician 
shall  have  made  these  problems  of  ours  their  own,  or  till  a  new 
Boscovich  shall  have  written  for  the  naturalist  the  new  Theoria 
Philosophiae  Naturalis. 

How  far,  even  then,  mathematics  will  suffice  to  describe,  and 
physics  to  explain,  the  fabric  of  the  body  no  man  can  foresee. 
It  may  be  that  all  the  laws  of  energy,  and  all  the  properties  of 
matter,  and  all  the  chemistry  of  all  the  colloids  are  as  powerless 
to  explain  the  body  as  they  are  impotent  to  comprehend  the 
soul.  For  my  part,  I  think  it  is  not  so.  Of  how  it  is  that  the 
soul  informs  the  body,  physical  science  teaches  me  nothing: 
consciousness  is  not  explained  to  my  comprehension  by  all  the 
nerve-paths  and  "neurones"  of  the  physiologist;  nor  do  I  ask  of 
physics  how  goodness  shines  in  one  man's  face,  and  evil  betrays 
itself  in  another.  But  of  the  construction  and  growth  and  working 
*  Whereby  he  incurred  the  reproach  of  Socrates,  in  the  Phaedo. 


I]  OF  DYNAMICAL  MORPHOLOGY  9 

of  the  body,  as  of  all  that  is  of  the  earth  earthy,  physical  science 
is,  in  my  humble  opinion,  our  only  teacher  and  guide*. 

Often  and  often  it  happens  that  our  physical  knowledge  is 
inadequate  to  explain  the  mechanical  working  of  the  organism ; 
the  phenomena  are  superlatively  complex,  the  procedure  is 
involved  and  entangled,  and  the  investigation  has  occupied  but 
a  few  short  lives  of  men.  When  physical  science  falls  short  of 
explaining  the  order  which  reigns  throughout  these  manifold 
phenomena, — an  order  more  characteristic  in  its  totality  than  any 
of  its  phenomena  in  themselves, — men  hasten  to  invoke  a  guiding 
principle,  an  entelechy,  or  call  it  what  you  will.  But  all  the  while, 
so  far  as  I  am  aware,  no  physical  law^,  any  more  than  that  of 
gravity  itself,  not  even  among  the  puzzles  of  chemical  "stereo- 
metry," or  of  physiological  "surface-action"  or  "osmosis,"  is 
known  to  be  transgressed  by  the  bodily  mechanism. 

Some  physicists  declare,  as  Maxwell  did,  that  atoms  or  mole- 
cules more  complicated  by  far  than  the  chemist's  hypotheses 
demand  are  requisite  to  explain  the  phenomena  of  life.  If  what 
is  implied  be  an  explanation  of  psychical  phenomena,  let  the 
point  be  granted  at  once ;  we  may  go  yet  further,  and  decline, 
with  Maxwell,  to  believe  that  anything  of  the  nature  of  physical 
complexity,  however  exalted,  could  ever  suffice.  Other  physicists, 
like  Auerbachf,  or  LarmorJ,  or  Joly§,  assure  us  that  our  laws  of 
thermodynamics  do  not  suffice,  or  are  "inappropriate,"  to  explain 
the  maintenance  or  (in  Joly's  phrase)  the  " accelerative  absorption" 

*  In  a  famous  lecture  (Conservation  of  Forces  applied  to  Organic  Nature, 
Proc.  Roy.  lustit.,  April  12,  1861),  Helmholtz  laid  it  down,  as  "the  fundamental 
principle  of  physiology,"  that  "There  may  be  other  agents  acting  in  the  living 
body  than  those  agents  which  act  in  the  inorganic  world;  but  those  forces,  as  far 
as  they  cause  chemical  and  mechanical  influence  in  the  body,  must»be  quite  of  the 
same  character  as  inorganic  forces :  in  this  at  least,  that  their  effects  must  be  ruled 
by  necessity,  and  must  always  be  the  same  when  acting  in  the  same  conditions; 
and  so  there  cannot  exist  any  arbitrary  choice  in  the  direction  of  their  actions." 
It  would  follow  from  this,  that,  like  the  other  "physical"  forces,  they  must  be 
subject  to  mathematical  anatysis  and  deduction.  Cf.  also  Dr  T.  Young's  Croonian 
Lecture  On  the  Heart  and  Arteries.  Phil.  Trans.  1809,  p.  1:  Coll.  Works,  i,  .511. 

f  Ektropismus,  oder  die  physikalische  Theorie  des  Lebens,  Leipzig,  1910. 

J  Wilde  Lecture,  Nature,  March  12,  1908;  ibid.  Sept.  6,  1900,  p.  485; 
Aether  and  Matter,  p.  288.     Cf.  also  Lord  Kelvin,  Fortnightly  Review,  1892,  p.  313. 

§  Joly,  The  Abundance  of  Life,  Proc.  Roy.  Dublin  Soc.  vii,  1890;  and  in 
/Scientific  Essays,  etc.  1915,  p.  60  et  seq. 


10  INTRODUCTORY  [ch, 

of  the  bodily  energies,  and  the  long  battle  against  the  cold  and 
darkness  which  is  death.  With  these  weighty  problems  I  am  not 
for  the  moment  concerned.  My  sole  purpose  is  to  correlate  with 
mathematical  statement  and  physical  law  certain  of  the  simpler 
outward  phenomena  of  organic  growth  and  structure  or  form: 
while  all  the  while  regarding,  ex  hypothesi,  for  the  purposes  of 
this  correlation,  the  fabric  of  the  organism  as  a  material  and 
mechanical   configuration. 

Physical  science  and  philosophy  stand  side  by  side,  and  one 
upholds  the  other.  Without  something  of  the  strength  of  physics, 
philosophy  would  be  weak  ;  and  without  something  of  philosophy's 
wealth,  physical  science  would  be  poor.  "  Rien  ne  retirera  du 
tissu  de  la  science  les  fils  d'or  que  la  main  du  philosophe  y  a 
introduits*."  But  there  are  fields  where  each,  for  a  while  at 
least,  must  work  alone ;  and  where  physical  science  reaches  its 
limitations,  physical  science  itself  must  help  us  to  discover. 
Meanwhile  the  appropriate  and  legitimate  postulate  of  the 
physicist,  in  approaching  the  physical  problems  of  the  body,  is 
that  with  these  physical  phenomena  no  ahen  influence  interferes. 
But  the  postulate,  though  it  is  certainly  legitimate,  and  though 
it  is  the  proper  and  necessary  prelude  to  scientific  enquiry,  may 
some  day  be  proven  to  be  untrue ;  and  its  disproof  will  not  be  to 
the  physicist's  confusion,  but  will  come  as  his  reward.  In  dealing 
with  forms  which  are  so  concomitant  with  life  that  they  are 
seemingly  controlled  by  hfe,  it  is  in  no  spirit  of  arrogant  assertive- 
ness  that  the  physicist  begins  his  argument,  after  the  fashion  of 
a  most  illustrious  exemplar,  with  the  old  formulary  of  scholastic 
challenge, — An  Vita  sit  ?     Dico  quod  non. 

The  terms  Form  and  Growth,  which  make  up  the  title  of  this 
little  book,  are  to  be  understood,  as  I  need  hardly  say,  in  their 
relation  to  the  science  of  organisms.  We  want  to  see  how,  in 
some  cases  at  least,  the  forms  of  living  things,  and  of  the  parts 
of  living  things,  can  be  explained  by  physical  considerations,  and 
to  reahse  that,  in  general,  no  organic  forms  exist  save  such  as 
are  in  conformity  with  ordinary  physical  laws.  And  while  growth 
is  a  somewhat  vague  word  for  a  complex  matter,  which  may 

*  Papillon,  Histoire  de  la  philosophie  moderne,  i,  p.  300. 


I]  OF  MATTER   AND   ENERGY  11 

depend  on  various  things,  from  simple  imbibition  of  water  to  the 
complicated  results  of  the  chemistry  of  nutrition,  it  deserves  to 
be  studied  in  relation  to  form,  whether  it  proceed  by  simple 
increase  of  size  without  obvious  alteration  of  form,  or  whether  it 
so  proceed  as  to  bring  about  a  gradual  change  of  form  and  the 
slow  development  of  a  more  or  less  complicated  structure. 

In  the  Newtonian  language  of  elementary  physics,  force  is 
recognised  by  its  action  in  producing  or  in  changing  motion,  or 
in  preventing  change  of  motion  or  in  maintaining  rest.  When  we 
deal  with  matter  in  the  concrete,  force  does  not,  strictly  speaking, 
enter  into  the  question,  for  force,  unlike  matter,  has  no  independent 
objective  existence.  It  is  energy  in  its  various  forms,  known  or 
unknown,  that  acts  upon  matter.  But  when  we  abstract  our 
thoughts  from  the  material  to  its  form,  or  from  the  thing  moved 
to  its  motions,  when  we  deal  with  the  subjective  conceptions  of 
form,  or  movement,  or  the  movements  that  change  of  form  implies, 
then  force  is  the  appropriate  term  for  our  conception  of  the  causes 
by  which  these  forms  and  changes  of  form  are  brought  about. 
When  we  use  the  term  force,  we  use  it,  as  the  physicist  always 
does,  for  the  sake  of  brevity,  using  a  symbol  for  the  magnitude 
and  direction  of  an  action  in  reference  to  the  symbol  or  diagram 
of  a  material  thing.  It  is  a  term  as  subjective  and  symbolic  as 
form  itself,  and  so  is  appropriately  to  be  used  in  connection 
therewith. 

The  form,  then,  of  any  portion  of  matter,  whether  it  be  living 
or  dead,  and  the  changes  of  form  that  are  apparent  in  its  movements 
and  in  its  growth,  may  in  all  cases  ahke  be  described  as  due  to 
the  action  of  force.  In  short,  the  form  of  an  object  is  a  "diagram 
of  forces,"  in  this  sense,  at  least,  that  from  it  we  can  judge  of  or 
deduce  the  forces  that  are  acting  or  have  acted  upon  it:  in  this 
strict  and  particular  sense,  it  is  a  diagram, — in  the  case  of  a  solid, 
of  the  forces  that  have  been  impressed  upon  it  when  its  conformation 
was  produced,  together  with  those  that  enable  it  to  retain  its 
conformation ;  in  the  case  of  a  Hquid  (or  of  a  gas)  of  the  forces  that 
^re  for  the  moment  acting  on  it  to  restrain  or  balance  its  own 
inherent  mobihty.  In  an  organism,  great  or  small,  it  is  not 
merely  the  nature  of  the  ynotions  of  the  living  substance  that  we 
must  interpret  in  terms  of  force  (according  to  kinetics),  but  also 


12   •  INTRODUCTORY  [ch. 

the  conformation  of  the  organism  itself,  whose  permanence  or 
equihbrium  is  explained  by  the  interaction  or  balance  of  forces, 
as  described  in  statics. 

If  we  look  at  the  living  cell  of  an  Amoeba  or  a  Spirogyra,  we 
see  a  something  which  exhibits  certain  active  movements,  and 
a  certain  fluctuating,  or  more  or  less  lasting,  form;  and  its  form 
at  a  given  moment,  just  like  its  motions,  is  to  be  investigated  by 
the  help  of  physical  methods,  and  explained  by  the  invocation  of 
the  mathematical  conception  of  force. 

Now  the  state,  including  the  shape  or  form,  of  a  portion  of 
matter,  is  the  resultant  of  a  number  of  forces,  which  represent  or 
symbolise  the  manifestations  of  various  kinds  of  energy;  and  it 
is  obvious,  accordingly,  that  a  great  part  of  physical  science  must 
be  understood  or  taken  for  granted  as  the  necessary  preliminary 
to  the  discussion  on  which  we  are  engaged.  But  we  may  at 
least  try  to  indicate,  very  briefly,  the  nature  of  the  principal 
forces  and  the  principal  properties  of  matter  with  which  our 
subject  obliges  us  to  deal.  Let  us  imagine,  for  instance,  the  case 
of  a  so-called  "simple"  organism,  such  as  Amoeba;  and  if  our 
short  list  of  its  physical  properties  and  conditions  be  helpful 
to  our  further  discussion,  we  need  not  consider  how  far  it 
be  complete  or  adequate  from  the  wider  physical  point  of 
view*. 

This  portion  of  matter,  then,  is  kept  together  by  the  inter- 
molecular  force  of  cohesion ;  in  the  movements  of  its  particles 
relatively  to  one  another,  and  in  its  own  movements  relative  to 
adjacent  matter,  it  meets  with  the  opposing  force  of  friction. 
It  is  acted  on  by  gravity,  and  this  force  tends  (though  slightly, 
owing  to  the  Amoeba's  small  mass,  and  to  the  small  difference 
between  its  density  and  that  of  the  surrounding  fluid),  to  flatten 
it  down  upon  the  sohd  substance  on  which  it  may  be  creeping. 
Our  Amoeba  tends,  in  the  next  place,  to  be  deformed  by  any 
pressure  from  outside,  even  though  sUght,  which  may  be  apphed 
to  it,  and  this  circumstance  shews  it  to  consist  of  matter  in  a 
fluid,  or  at  least  semi-fluid,  state :  which  state  is  further  indicated 
when  we  observe  streaming  or  current  motions  in  its  interior. 

*  With  the  special  and  important  properties  of  colloidal  matter  we  are,  for 
the  time  being,  not  concerned. 


I]  OF  MATTER   AND   ENERGY  13 

Like  other  fluid  bodies,  its  surface,  whatsoever  other  substance, 
gas,  Hquid  or  sohd,  it  be  in  contact  with,  and  in  varying  degree 
according  to  the  nature  of  that  adjacent  substance,  is  the  seat 
of  molecular  force  exhibiting  itself  as  a  surface-tension,  from  the 
action  of  which  many  important  consequences  follow,  which 
greatly  afEect  the  form  of  the  fluid  surface. 

While  the  protoplasm  of  the  Amoeba  reacts  to  the  slightest 
pressure,  and  tends  to  "flow,"  and  while  we  therefore  speak  of  it 
as  a  fluid,  it  is  evidently  far  less  mobile  than  such  a  fluid,  for 
instance,  as  water,  but  is  rather  like  treacle  in  its  slow  creeping 
movements  as  it  changes  its  shape  in  response  to  force.  Such 
fluids  are  said  to  have  a  high  viscosity,  and  this  viscosity  obviously 
acts  in  the  way  of  retarding  change  of  form,  or  in  other  words 
of  retarding  the  effects  of  any  disturbing  action  of  force.  When 
the  viscous  fluid  is  capable  of  being  drawn  out  into  fine  threads, 
a  property  in  which  we  know  that  the  material  of  some  Amoebae 
differs  greatly  from  that  of  others,  we  say  that  the  fluid  is  also 
viscid,  or  exhibits  viscidity.  Again,  not  by  virtue  of  our  Amoeba 
being  liquid,  but  at  the  same  time  in  vastly  greater  measure  than  if  it 
were  a  solid  (though  far  less  rapidly  than  if  it  were  a  gas),  a  process 
of  molecular  diffusion  is  constantly  going  on  within  its  substance, 
by  which  its  particles  interchange  their  places  within  the  mass, 
while  surrounding  fluids,  gases  and  sohds  in  solution  diffuse  into 
and  out  of  it.  In  so  far  as  the  outer  wall  of  the  cell  is  different 
in  character  from  the  interior,  whether  it  be  a  mere  pellicle  as 
in  Amoeba  or  a  firm  cell-wall  as  in  Protococcus,  the  diffusion 
which  takes  place  through  this  wall  is  sometimes  distinguished 
under  the  term  osmosis. 

Within  the  cell,  chemical  forces  are  at  work,  and  so  also  in 
all  probability  (to  judge  by  analogy)  are  electrical  forces ;  and 
the  organism  reacts  also  to  forces  from  without,  that  have  their 
origin  in  chemical,  electrical  and  thermal  influences.  The  pro- 
cesses of  diffusion  and  of  chemical  activity  within  the  cell  result, 
by  the  drawing  in  of  water,  salts,  and  food-material  with  or  without 
chemical  transformation  into  protoplasm,  in  growth,  and  this 
complex  phenomenon  we  shall  usually,  without  discussing  its 
nature  and  origin,  describe  and  picture  as  a  force.  Indeed  we 
shall  manifestly  be  inclined  to  use  the  term  growth  in  two  senses. 


14  INTRODUCTORY  [ch. 

just  indeed  as  we  do  in  the  case  of  attraction  or  gravitation, 
on  the  one  hand  as  a  process,  and  on  the  other  hand  as  a 
force. 

In  the  phenomena  of  cell-division,  in  the  attractions  or  repul- 
sions of  the  parts  of  the  dividing  nucleus  and  in  the  "  caryokinetic  " 
figures  that  appear  in  connection  with  it,  we  seem  to  see  in  opera- 
tion forces  and  the  effects  of  forces,  that  have,  to  say  the  least  of 
it,  a  close  analogy  with  known  physical  phenomena ;  and  to  this 
matter  we  shall  afterwards  recur.  But  though  they  resemble 
known  physical  phenomena,  their  nature  is  still  the  subject  of 
much  discussion,  and  neither  the  forms  produced  nor  the  forces 
at  work  can  yet  be  satisfactorily  and  simply  explained.  We  may 
readily  admit,  then,  that  besides  phenomena  which  are  obviously 
physical  in  their  nature,  there  are  actions  visible  as  well  as 
invisible  taking  place  within  living  cells  which  our  knowledge 
does  not  permit  us  to  ascribe  with  certainty  to  any  known  physical 
force ;  and  it  may  or  may  not  be  that  these  phenomena  will  yield 
in  time  to  the  methods  of  physical  investigation.  Whether  or 
no,  it  is  plain  that  we  have  no  clear  rule  or  guide  as  to  what  is 
"vital"  and  what  is  not;  the  whole  assemblage  of  so-called  vital 
phenomena,  or  properties  of  the  organism,  cannot  be  clearly 
classified  into  those  that  are  physical  in  origin  and  those  that  are 
sui  generis  and  peculiar  to  living  things.  All  we  can  do  meanwhile 
is  to  analyse,  bit  by  bit,  those  parts  of  the  whole  to  which  the 
ordinary  laws  of  the  physical  forces  more  or  less  obviously  and 
clearly  and  indubitably  apply. 

Morphology  then  is  not  only  a  study  of  material  things  and 
of  the  forms  of  material  things,  but  has  its  dynamical  aspect, 
under  which  we  deal  with  the  interpretation,  in  terms  of  force, 
of  the  operations  of  Energy.  And  here  it  is  well  worth  while 
to  remark  that,  in  dealing  wdth  the  facts  of  embryology  or  the 
phenomena  of  inheritance,  the  common  language  of  the  books 
seems  to  deal  too  much  with  the  material  elements  concerned,  as 
the  causes  of  development,  of  variation  or  of  hereditary  trans- 
mission. Matter  as  such  produces  nothing,  changes  nothing,  does 
nothing ;  and  however  convenient  it  may  afterwards  be  to  abbre- 
viate our  nomenclature  and  our  descriptions,  we  must  most 
carefully  reahse  in  the  outset  that  the  spermatozoon,  the  nucleus, 


I]  OF   MATTER   AND   ENERGY  15 

the  chromosomes  or  the  germ-plasm  can  never  act  as  matter 
alone,  but  only  as  seats  of  energy  and  as  centres  of  force.  And 
this  is  but  an  adaptation  (in  the  light,  or  rather  in  the  con- 
ventional symbolism,  of  modern  physical  science)  of  the  old 
saying  of  the  philosopher:    apxv  l^P  V  4>^o-i^  /uLaWov  r?}?  vXrjq. 


CHAPTER   II 

ON  MAGNITUDE 

To  terms  of  magnitude,  and  of  direction,  must  we  refer  all 
our  conceptions  of  form.  For  the  form  of  an  object  is  defined 
when  '  we  know  its  magnitude,  actual  or  relative,  in  various 
directions ;  and  growth  involves  the  same  conceptions  of  magnitude 
and  direction,  with  this  addition,  that  they  are  supposed  to  alter 
in  time.  Before  we  proceed  to  the  consideration  of  specific  form, 
it  will  be  worth  our  while  to  consider,  for  a  little  while,  certain 
phenomena  of  spatial  magnitude,  or  of  the  extension  of  a  body 
in  the  several  dimensions  of  space*. 

We  are  taught  by  elementary  mathematics  that,  in  similar 
solid  figures,  the  surface  increases  as  the  square,  and  the  volume 
as  the  cube,  of  the  linear  dimensions.  If  we  take  the  simple  case 
of  a  sphere,  with  radius  r,  the  area  of  its  surface  is  equal  to  477/^, 
and  its  volume  to  47rf^;    from  which  it  follows  that  the  ratio  of 

o 

volume  to  surface,  or  V/S,  is  ^r.  In  other  words,  the  greater  the 
radius  (or  the  larger  the  sphere)  the  greater  will  be  its  volume,  or 
its  mass  (if  it  be  uniformly  dense  throughout),  in  comparison  with 
its  superficial  area.  And,  taking  L  to  represent  any  linear  dimen- 
sion, we  may  write  the  general  equations  in  the  form 

SocL^     F  oc  L\ 

or  S^k.L^   and   V  =  k'.L^; 

and  (S^  X  L. 

From  these  elementary  principles  a  great  number  of  conse- 
quences follow,  all  more  or  less  interesting,  and  some  of  them  of 
great  importance.     In  the  first  place,  though  growth  in  length  (let 

*  Cf.    Hans    Przibrain,    Anwendung  elementarer    Mathematik    auf   Biologische 
Probleme  (in  Roux's  Vortrdge,  Heft  ra),  Leipzig,  1908,  p.  10. 


CH.  II]  OF   SURFACE   AND   VOLUME  17 

us  say)  and  growth  in  volume  (which  is  usually  tantamount  to 
mass  or  weight)  are  parts  of  one  and  the  same  process  or  pheno- 
menon, the  one  attracts  our  attention  by  its  increase,  very  much 
more  than  the  other.  For  instance  a  fish,  in  doubhng  its  length, 
multipHes  its  weight  by  no  less  than  eight  times ;  and  it  all  but 
doubles  its  weight  in  growing  from  four  inches  long  to  five. 

In  the  second  place  we  see  that  a  knowledge  of  the  correlation 
between  length  and  weight  in  any  particular  species  of  animal, 
in  other  words  a  determination  of  k  in  the  formula  W  ^  k  .  L^, 
enables  us  at  any  time  to  translate  the  one  magnitude  into  the 
other,  and  (so  to  speak)  to  weigh  the  animal  with  a  measuring- 
rod;  this  however  being  always  subject  to  the  condition  that  the 
animal  shall  in  no  way  have  altered  its  form,  nor  its  specific 
gravity.  That  its  specific  gravity  or  density  should  materially  or 
rapidly  alter  is  not  very  hkely;  but  as  long  as  growth  lasts, 
changes  of  form,  even  though  inappreciable  to  the  eye,  are  likely 
to  go  on.  Now  weighing  is  a  far  easier  and  far  more  accurate 
operation  than  measuring;  and  the  measurements  which  would 
reveal  shght  and  otherwise  imperceptible  changes  in  the  form  of 
a  fish — shght  relative  difiierences  between  length,  breadth  and 
depth,  for  instance, — would  need  to  be  very  dehcate  indeed.  But 
if  we  can  make  fairly  accurate  determinations  of  the  length, 
which  is  very  much  the  easiest  dimension  to  measure,  and  then 
correlate  it  with  the  weight,  then  the  value  of  k,  according  to 
whether  it  varies  or  remains  constant,  will  tell  us  at  once  whether 
there  has  or  has  not  been  a  tendency  to  gradual  alteration  in  the 
general  form.  To  this  subject  we  shall  return,  when  we  come  to 
consider  more  particularly  the  rate  of  growth. 

But  a  much  deeper  interest  arises  out  of  this  changing  ratio 
of  dimensions  when  we  come  to  consider  the  inevitable  changes 
of  physical  relations  with  which  it  is  bound  up.  We  are  apt,  and 
even  accustomed,  to  think  that  magnitude  is  so  purely  relative 
that  differences  of  magnitude  make  no  other  or  more  essential 
difference ;  that  Lilhput  and  Brobdingnag  are  all  ahke,  according 
as  we  look  at  them  through  one  end  of  the  glass  or  the  other. 
But  this  is  by  no  means  so;  for  scale  has  a  very  marked  effect 
upon  physical  phenomena,  and  the  effect  of  scale  constitutes  what 
is  known  as  the  principle  of  similitude,  or  of  dynamical  similarity. 


18  ON  MAGNITUDE  [ch. 

This  effect  of  scale  is  simply  due  to  the  fact  that,  of  the  physical 
forces,  some  act  either  directly  at  the  surface  of  a  body,  or  other- 
wise in  proportion  to  the  area  of  surface;  and  others,  such  as 
gravity,  act  on  all  particles,  internal  and  external  ahke,  and  exert 
a  force  which  is  proportional  to  the  mass,  and  so  usually  to  the 
volume,  of  the  body. 

The  strength  of  an  iron  girder  obviously  varies  with  the 
cross-section  of  its  members,  and  each  cross-section  varies  as  the 
square  of  a  linear  dimension ;  but  the  weight  of  the  whole  structure 
varies  as  the  cube  of  its  linear  dimensions.  And  it  follows  at  once 
that,  if  we  build  two  bridges  geometrically  similar,  the  larger  is 
the  weaker  of  the  two  *.  It  was  elementary  engineering  experience 
such  as  this  that  led  Herbert  Spencer  f  to  apply  the  principle  of 
simihtude  to  biology. 

The  same  principle  had  been  admirably  applied,  in  a  few  clear 
instances,  by  LesageJ,  a  celebrated  eighteenth  century  physician 
of  Geneva,  in  an  unfinished  and  unpublished  work§.  Lesage 
argued,  for  mstance,  that  the  larger  ratio  of  surface  to  mass  would 
lead  in  a  small  animal  to  excessive  transpiration,  were  the  skin 
as  "porous"  as  our  own;  and  that  we  may  hence  account  for 
the  hardened  or  thickened  skins  of  insects  and  other  small  terrestrial 
animals.  Again,  since  the  weight  of  a  fruit  increases  as  the  cube 
of  its  dimensions,  while  the  strength  of  the  stalk  increases  as  the 
square,  it  follows  that  the  stalk  should  grow  out  of  apparent  due 
proportion  to  the  fruit;  or  alternatively,  that  tall  trees  should 
not  bear  large  fruit  on  slender  branches,  and  that  melons  and 
pumpkins  must  lie  upon  the  ground.  And  again,  that  in  quad- 
rupeds a  large  head  must  be  supported  on  a  neck  which  is  either 

*  The  subject  is  treated  from  an  engineering  j^oint  of  view  by  Prof.  James 
Thomson,  Comparisons  of  Similar  Structures  as  to  Elasticity,  Strength,  and 
StabUity,  Trans.  Inst.  Engineers,  Scotland,  1876  {Collected  Papers,  1912,  pp.  361- 
372),  and  by  Prof.  A.  Barr,  ibid.  1899;    see  also  Rayleigh,  Nattire,  April  22,  1915. 

t  Cf.  Spencer,  The  Form  of  the  Earth,  etc.,  Phil.  Mag.  xxx,  pp.  194-6, 
1847;    also  Principles  of  Biology,  pt.  n,  ch.  i,  1864  (p.   123,  etc.). 

%  George  Louis  Lesage  (1724-1803),  well  known  as  the  author  of  one  of  the  few 
attempts  to  explam  gravitation.  (Cf.  Leray,  Constitution  de  la  Matiere,  1869; 
Kelvin,  Proc.  R.  S.  E.  vn,  p.  577,  1872,  etc. ;  Clerk  Maxwell,  Phil.  Trans,  vol.  157, 
p.  50,  1867;  art.  "Atom,"  Emycl.  Brit.  1875,  p.  46.) 

§  Cf.  Pierre  Prevost,  Noticts  de  la  vie  et  des  ecrits  de  Lesage,  1805;  quoted  by 
Janet,  Causes  Finales,  app.  in. 


II]  THE   PRINCIPLE   OF   SIMILITUDE  19 

excessively  thick  and  strong,  like  a  bull's,  or  very  short  like  the 
neck  of  an  elephant. 

But  it  was  Galileo  who,  wellnigh  300  years  ago,  had  first  laid 
down  this  general  principle  which  we  now  know  by  the  name  of  the 
principle  of  siniihtude ;  and  he  did  so  with  the  utmost  possible 
clearness,  and  with  a  great  wealth  of  illustration,  drawn  from 
structures  hving  and  dead*.  He  showed  that  neither  can  man 
build  a  house  nor  can  nature  construct  an  animal  beyond  a  certain 
size,  while  retaining  the  same  proportions  and  employing  the 
same  materials  as  sufiiced  in  the  case  of  a  smaller  structure  f. 
The  thing  will  fall  to  pieces  of  its  own  weight  unless  we  either 
change  its  relative  proportions,  which  will  at  length  cause  it  to 
become  clumsy,  monstrous  and  inefficient,  or  else  we  must  find 
a  new  material,  harder  and  stronger  than  was  used  before.  Both 
processes  are  famihar  to  us  in  nature  and  in  art,  and  practical 
appUcations,  undreamed  of  by  Gahleo,  meet  us  at  every  turn  in 
this  modern  age  of  steel. 

Again,  as  Gahleo  was  also  careful  to  explain,  besides  the 
questions  of  pure  stress  and  strain,  of  the  strength  of  muscles  to 
lift  an  increasing  weight  or  of  bones  to  resist  its  crushing  stress, 
we  have  the  very  important  question  of  bending  moments.  This 
question  enters,  more  or  less,  into  our  whole  range  of  problems ; 
it  afiects,  as  we  shall  afterwards  see,  or  even  determines  the  whole 
form  of  the  skeleton,  and  is  very  important  in  such  a  case  as  that 
of  a  tall  tree  J. 

Here  we  have  to  determine  the  point  at  which  the  tree  will 
curve  under  its  own  weight,  if  it  be  ever  so  little  displaced  from 
the  perpendicular §.     In  such  an  investigation  we  have  to  make 

*  Discorsi  e  Dimostrazioni  majematiche,  intorno  a  due  nuove  scienze. 
attenenti  alia  Mecanica,  ed  ai  Movimenti  Local! :  appresso  gli  Elzevirii,  mdcxxxviii. 
Opere,  ed.  Favaro,  vni,  p.  169  seq.  Transl.  by  Henry  Crew  and  A.  de  Salvio, 
1914,  p.  130,  etc.     See  Nature,  June  17,  191.5. 

t  So  Werner  remarked  that  Michael  Angelo  and  Bramanti  could  not  have  built 
of  gypsum  at  Paris  on  the  scale  they  built  of  travertin  in  Rome. 

X  Sir  G.  Greenhill,  Determination  of  the  greatest  height  to  which  a  Tree  of 
given  proportions  can  grow,  Cambr.  Phil.  Soc.  Pr.  iv,  p.  65,  1881,  and  Chree, 
ibid.  VII,  1892.     Cf.  Pojoiting  and  Thomson's  Properties  of  Matter,  1907,  p  99. 

§  In  like  manner  the  wheat-straw  bends  over  under  the  weight  of  the  loaded 
ear,  and  the  tip  of  the  cat's  tail  bends  over  when  held  upright,— not  because  they 
"possess  flexibility,"  but  because  they  outstrip  the  dimensions  withm  which  stable 

2—2 


20  ON  MAGNITUDE  [ch. 

some  assumptions, — for  instance,  with  regard  to  the  trunk,  that 
it  tapers  uniformly,  and  with  regard  to  the  branches  that  their 
sectional  area  varies  according  to  some  definite  law,  or  (as  Ruskin 
assumed*)  tends  to  be  constant  in  any  horizontal  plane;  and  the 
mathematical  treatment  is  apt  to  be  somewhat  difficult.  But 
Greenhill  has  shewn  that  (on  such  assumptions  as  the  above), 
a  certain  British  Columbian  pine-tree,  which  yielded  the  Kew  flag- 
staff measuring  22f  ft.  in  height  with  a  diameter  at  the  base  of 

21  inches,  could  not  possibly,  by  theory,  have  grown  to  more 
than  about  300  ft.  It  is  very  curious  that  Galileo  suggested 
precisely  the  same  height  {dugento  braccia  alta)  as  the  utmost 
limit  of  the  growth  of  a  tree.  In  general,  as  Greenhill  shews,  the 
diameter  of  a  homogeneous  body  must  increase  as  the  power  3/2 
of  the  height,  which  accounts  for  the  slender  proportions  of  young 
trees,  compared  with  the  stunted  appearance  of  old  and  large 
ones|.  In  short,  as  Goethe  says  in  Wahrheit  und  Dichtung,  "Es 
ist  dafiir  gesorgt  dass  die  Baume  nicht  in  den  Himmel  wachsen." 
But  Eiffel's  great  tree  of  steel  (1000  feet  high)  is  built  to  a 
very  differeiit  plan;  for  here  the  profile  of  the  tower  follows  the 
logarithmic  curve,  giving  equal  strength  throughout,  according 
to  a  principle  which  we  shall  have  occasion  to  discuss  when  we 
come  to  treat  of  "form  and  mechanical  efficiency"  in  connection 
with  the  skeletons  of  animals. 

Among  animals,  we  may  see  in  a  general  way,  without  the  help 
of  mathematics  or  of  physics,  that  exaggerated  bulk  brings  with 
it  a  certain  clumsiness,  a  certain  inefiiciency,  a  new  element  of 
risk  and  hazard,  a  vague  preponderance  of  disadvantage.  The 
case  was  well  put  by  Owen,  in  a  passage  which  has  an  interest 
of  its  own  as  a  premonition  (somewhat  like  De  Candolle's)  of  the 
"struggle  for  existence."  Owen  wrote  as  follows $:  "In  pro- 
portion to  the  bulk  of  a  species  is  the  difficulty  of  the  contest 
which,  as  a  living  organised  whole,  the  individual  of  such  species 

equilibrium  is  possible  in  a  vertical  position.  The  kitten's  tail,  on  the  other  hand, 
stands  up  spiky  and  straight. 

*  Modern  Painters. 

t  The  stem  of  the  giant  bamboo  may  attain  a  height  of  60  metres,  while  not 
more  than  about  40  cm.  in  diameter  near  its  base,  which  dimensions  are  not  very 
far  short  of  the  theoretical  limits  (A.  J.  Ewart,  Phil.  Trans,  vol.  198,  p.  71,  1906). 

X  Trans.  Zool.  Soc.  iv,  1850,  p.  27. 


II]  THE   PRINCIPLE   OF  SIMILITUDE  21 

has  to  maintain  against  the  surrounding  agencies  that  are  ever 
tending  to  dissolve  the  vital  bond,  and  subjugate  the  living 
matter  to  the  ordinary  chemical  and  physical  forces.  Any 
changes,  therefore,  in  such  external  conditions  as  a  species  may 
have  been  originally  adapted  to  exist  in,  will  militate  against  that 
existence  in  a  degree  proportionate,  perhaps  in  a  geometrical  ratio, 
to  the  bulk  of  the  species.  If  a  dry  season  be  greatly  prolonged, 
the  large  mammal  will  suffer  from  the  drought  sooner  than  the 
small  one ;  if  any  alteration  of  climate  affect  the  quantity  of 
vegetable  food,  the  bulky  Herbivore  will  first  feel  the  effects  of 
stinted  nourishment." 

But  the  principle  of  Galileo  carries  us  much  further  and  along 
more  certain  lines. 

The  tensile  strength  of  a  muscle,  like  that  of  a  rope  or  of  our 
girder,  varies  with  its  cross-section ;  and  the  resistance  of  a  bone 
to  a  crushing  stress  varies,  again  like  our  girder,  with  its  cross- 
section.  But  in  a  terrestrial  animal  the  weight  which  tends  to 
crush  its  limbs  or  which  its  muscles  have  to  move,  varies  as  the 
cube  of  its  linear  dimensions ;  and  so,  to  the  possible  magnitude 
of  an  animal,  living  under  the  direct  action  of  gravity,  there  is 
a  definite  limit  set.  The  elephant,  in  the  dimensions  of  its  limb- 
bones,  is  already  shewing  signs  of  a  tendency  to  disproportionate 
thickness  as  compared  with  the  smaller  mammals ;  its  movements 
are  in  many  ways  hampered  and  its  agility  diminished :  it  is 
already  tending  towards  the  maximal  limit  of  size  which  the 
physical  forces  permit.  But,  as  Galileo  also  saw,  if  the  animal 
be  wholly  immersed  in  water,  like  the  whale,  (or  if  it  be  partly 
so,  as  was  in  all  probability  the  case  with  the  giant  reptiles  of  our 
secondary  rocks),  then  the  weight  is  counterpoised  to  the  extent 
of  an  equivalent  volume  of  water,  and  is  completely  counterpoised 
if  the  density  of  the  animal's  body,  with  the  included  air,  be 
identical  (as  in  a  whale  it  very  nearly  is)  with  the  water  around. 
Under  these  circumstances  there  is  no  longer  a  physical  barrier 
to  the  indefinite  growth  in  magnitude  of  the  animal*.     Indeed, 

*  It  would  seem  to  be  a  common  if  not  a  general  rule  that  marine  organisms, 
zoophytes,  molluscs,  etc.,  tend  to  be  larger  than  the  corresponding  and  closely- 
related  forms  living  in  fresh  water.  While  the  phenomenon  may  have  various 
causes,  it  has  been  attributed  (among  others)  to  the  simple  fact  that  the  forces  of 
growth  are  less  antagonised  by  gravity  in  the  denser  medium  (cf.  Houssay,  La 


22  ON  MAGNITUDE  [ch. 

in  the  case  of  the  aquatic  animal  there  is,  as  Spencer  pointed  out, 
a  distinct  advantage,  in  that  the  larger  it  grows  the  greater  is 
its  velocity.  For  its  available  energy  depends  on  the  mass  of 
its  muscles ;  while  its  motion  through  the  water  is  opposed,  not 
by  gravity,  but  by  "skin-friction,"  which  increases  only  as  the 
square  of  its  dimensions ;  all  other  things  being  equal,  the  bigger 
the  ship,  or  the  bigger  the  fish,  the  faster  it  tends  to  go,  but  only 
in  the  ratio  of  the  square  root  of  the  increasing  length.  For  the 
mechanical  work  {W)  of  which  the  fish  is  capable  being  pro- 
portional to  the  mass  of  its  muscles,  or  the  cube  of  its  linear 
dimensions :  and  again  this  work  being  wholly  done  in  producing 
a  velocity  (F)  against  a  resistance  {R)  which  increases  as  the 
square  of  the  said  linear  dimensions ;    we  have  at  once 

W  =  l\ 

and  also  W  =  RV'- =  l^VK 

Therefore  P  =  IW\    and    V  =  y/l. 

This  is  what  is  known  as  Fronde's  Law  of  the  correspondence  of 
speeds. 

But  there  is  often  another  side  to  these  questions,  which  makes 
them  too  complicated  to  answer  in  a  word.  For  instance,  the 
work  (per  stroke)  of  which  two  similar  engines  are  capable  should 
obviously  vary  as  the  cubes  of  their  linear  dimensions,  for  it 
varies  on  the  one  hand  with  the  surface  of  the  piston,  and  on  the 
other,  with  the  length  of  the  stroke ;  so  is  it  likewise  in  the  animal, 
where  the  corresponding  variation  depends  on  the  cross-section  of 
the  muscle,  and  on  the  space  through  which  it  contracts.  But 
in  two  precisely  similar  engines,  the  actual  available  horse-power 
varies  as  the  square  of  the  linear  dimensions,  and  not  as  the 
cube;  and  this  for  the  obvious  reason  that  the  actual  energy 
developed  depends  upon  the  heating-surface  of  the  boiler*.  So 
likewise  must  there  be  a  similar  tendency,  among  animals,  for  the 
rate  of  supply  of  kinetic  energy  to  vary  with  the  surface  of  the 

Forme  et  la  Vie,  1900,  p.  815).  The  effect  of  gravity  on  outward  form  is 
illustrated,  for  instance,  by  the  contrast  between  the  uniformly  upward  branching 
of  a  sea-weed  and  the  drooping  curves  of  a  shrub  or  tree. 

*  The  analogy  is  not  a  very  strict  one.  We  are  not  taking  account,  for  instance, 
of  a  proportionate  increase  in  thickness  of  the  boiler-plates. 


II]  OF  FROUDE'S  LAW  23 

lung,  that  is  to  say  (other  things  being  equal)  with  the  square  of 
the  linear  dimensions  of  the  animal.  We  may  of  course  (departing 
from  the  condition  of  similarity)  increase  the  heating-surface  of 
the  boiler,  by  means  of  an  internal  system  of  tubes,  without 
increasing  its  outward  dimensions,  and  in  this  very  way  nature 
increases  the  respiratory  surface  of  a  lung  by  a  complex  system 
of  branching  tubes  and  minute  air-cells ;  but  nevertheless  in 
two  similar  and  closely  related  animals,  as  also  in  two  steam- 
engines  of  precisely  the  same  make,  the  law  is  bound  to  hold  that 
the  rate  of  working  must  tend  to  vary  with  the  square  of  the 
linear  dimensions,  according  to  Froude's  law  of  steamshij)  com- 
parison. In  the  case  of  a  very  large  ship,  built  for  speed,  the 
difficulty  is  got  over  by  increasing  the  size  and  number  of  the 
boilers,  till  the  ratio  between  boiler-room  and  engine-room  is 
far  beyond  what  is  required  in  an  ordinary  small  vessel  * ;  but 
though  we  find  lung-space*  increased  among  animals  where 
greater  rate  of  working  is  required,  as  in  general  among  birds, 
I  do  not  know  that  it  can  be  shewn  to  increase,  as  in  the 
"  over-boilered "  ship,  with  the  size  of  the  animal,  and  in  a  ratio 
which  outstrips  that  of  the  other  bodily  dimensions.  If  it  be  the 
case  then,  that  the  working  mechanism  of  the  muscles  should  be 
able  to  exert  a  force  proportionate  to  the  cube  of  the  linear 
bodily  dimensions,  while  the  respiratory  mechanism  can  only 
supply  a  store  of  energy  at  a  rate  proportional  to  the  square  of 
the  said  dimensions,  the  singular  result  ought  to  follow  that,  in 
swimming  for  instance,  the  larger  fish  ought  to  be  able  to  put  on 
a  spurt  of  speed  far  in  excess  of  the  smaller  one ;  but  the  distance 
travelled  by  the  year's  end  should  be  very  much  alike  for  both 
of  them.     And  it  should  also  follow  that  the  curve  of  fatigue 

*  Let  L  be  the  length,  S  the  (wetted)  surface,  T  the  tonnage,  D  the  displacement 
(oi"  volume)  of  a  ship;  and  let  it  cross  the  Atlantic  at  a  speed  V.  Then,  in  com- 
paring two  ships,  similarly  constructed  but  of  different  magnitudes,  we  know  that 
L  =  V",  S  =  L~  =  V*,  D  =  T=L^  =  V«;  also  R  (resistance)  =  S  .  F-  =  F« ;  H  (horse- 
power) =  R  .V  ^V ;  and  the  coal  (C)  necessary  for  the  voyage  =  HjV  =  F*.  That 
is  to  say,  in  ordinary  engineering  language,  to  increase  the  speed  across  the  Atlantic 
by  1  per  cent,  the  ship's  length  must  be  increased  2  per  cent.,  her  tonnage  or 
displacement  6  per  cent.,  her  coal-consumpt  also  6  per  cent.,  her  horse- power, 
and  therefore  her  boiler-capacity,  7  per  cent.  Her  bunkers,  accordingly,  keep 
pace  with  the  enlargement  of  the  ship,  but  her  boilers  tend  to  increase  out  of 
proportion  to  the  space  available. 


24  ON  MAGNITUDE  [ch. 

should  be  a  steeper  one,  and  the  staying  power  should  be  less,  in 
the  smaller  than  in  the  larger  individual.  This  is  the  case  of  long- 
distance racing,  where  the  big  winner  puts  on  his  big  spurt  at  the 
end.  And  for  an  analogous  reason,  wise  men  know  that  in  the 
'Varsity  boat-race  it  is  judicious  and  prudent  to  bet  on  the  heavier 
crew. 

Leaving  aside  the  question  of  the  supply  of  energy,  and  keeping 
to  that  of  the  mechanical  efficiency  of  the  machine,  we  may  find 
endless  biological  illustrations  of  the  principle  of  simihtude. 

In  the  case  of  the  flying  bird  (apart  from  the  initial  difficulty  of 
raising  itself  into  the  air,  which  involves  another  problem)  it  may 
be  shewn  that  the  bigger  it  gets  (all  its  proportions  remaining  the 
same)  the  more  difficult  it  is  for  it  to  maintain  itself  aloft  in  flight. 
The  argument  is  as  follows : 

In  order  to  keep  aloft,  the  bird  must  communicate  to  the  air 
a  downward  momentum  equivalent- to  its  own  weight,  and  there- 
fore proportional  to  tlie  cube  of  its  own  linear  dimensions.  But 
the  momentum  so  communicated  is  proportional  to  the  mass  of 
air  driven  downwards,  and  to  the  rate  at  which  it  is  driven :  the 
mass  being  proportional  to  the  bird's  wing-area,  and  also  (with 
any  given  slope  of  wing)  to  the  speed  of  the  bird,  and  the  rate 
being  again  proportional  to  the  bird's  speed;  accordingly  the 
whole  momentum  varies  as  the  wing-area,  i.e.  as  the  square  of  the 
linear  dimensions,  and  also  as  the  square  of  the  speed.  Therefore, 
in  order  that  the  bird  may  maintain  level  flight,  its  speed  must 
be  proportional  to  the  square  root  of  its  linear  dimensions. 

Now  the  rate  at  which  the  bird,  in  steady  ffight,  has  to  work 
in  order  to  drive  itself  forward,  is  the  rate  at  which  it  commmiicates 
energy  to  the  air;  and  this  is  proportional  to  mV^,  i.e.  to  the 
mass  and  to  the  square  of  the  velocity  of  the  air  displaced.  But 
the  mass  of  air  displaced  per  second  is  proportional  to  the  wing- 
area  and  to  the  speed  of  the  bird's  motion,  and  therefore  to  the 
power  2|  of  the  hnear  dimensions ;  and  the  speed  at  which  it 
is  displaced  is  proportional  to  the  bird's  speed,  and  therefore  to 
the  square  root  of  the  hnear  dimensions.  Therefore  the  energy 
communicated  per  second  (being  proportional  to  the  mass  and  to 
the  square  of  the  speed)  is  jointly  proportional  to  the  power  2|  of 
the  linear  dimensions,  as  above,  and  to  the  first  power  thereof : 


II]  THE   PRINCIPLE   OF   SIMILITUDE  25 

that  is  to  say,  it  increases  in  proportion  to  the  power  3|  of  the 
linear  dimensions,  and  therefore  faster  than  the  weight  of  the 
bird  increases. 

Put  in  mathematical  form,  the  equations  are  as  follows : 

{m  =  the  mass  of  air  thrust  downwards ;  F  its  velocity, 
proportional  to  that  of  the  bird;  M  its  momentum;  I  a  Hnear 
dimension  of  the  bird ;  w  its  weight ;  W  the  work  done  in  moving 
itself  forward.) 

M  =  w  =  l^. 

But  M^mV,    and    7n  =  PV. 

Therefore  M  ^  l^  V^ 

and        ?2  y2  _  jz^ 

or  7  =  V?- 

But,  again,  W  =  mV^  ^PV  x  V^ 

=  PxVl^l 

The  work  requiring  to  be  done,  then,  varies  as  the  power  3|  of 
the  bird's  linear  dimensions,  while  the  work  of  which  the  bird  is 
capable  depends  on  the  mass  of  its  muscles,  and  therefore  varies 
as  the  cube  of  its  hnear  dimensions*.  The  disproportion  does  not 
seem  at  first  sight  very  great,  but  it  is  quite  enough  to  tell.  It  is 
as  much  as  to  say  that,  every  time  we  double  the  hnear  dimensions 
of  the  bird,  the  difficulty  of  flight  is  increased  in  the  ratio  of 
2^  :  2^-,  or  8  :  11-3,  or,  say,  1:1-4.  If  we  take  the  ostrich  to 
exceed  the  sparrow  in  linear  dimensions  as  25  :  1,  which  seems  well 
within  the  mark,  we  have  the  ratio  between  25^*  and  25^,  or 
between  5'^:  5^;  in  other  words,  flight  is  just  five  times  more 
difficult  for  the  larger  than  for  the  smaller  birdf. 

The  above  investigation  includes,  besides  the  final  result,  a 
number  of  others,  explicit  or  implied,  which  are  of  not  less  im- 
portance.    Of  these  the  simplest  and  also  the  most  important  is 

*  This  is  the  result  ari'ived  at  by  Helmholtz,  Ueber  ein  Theorem  geometrisch 
ahnliche  Bewegungen  fliissiger  Korper  betreffend,  nebst  Anwendung  auf  das 
Problem  Luftballons  zu  lenken,  Monatsber.  Akad.  Berlin,  1873,  pp.  501-14.  It 
was  criticised  and  challenged  (somewhat  rashly)  by  K.  Miillenhof,  Die  Grosse 
der  Flugflachen,  etc.,  Pfluger''s  Archiv,  xxxv,  p.  407,  xxxvi,  p.  548,  1885. 

t  Cf.  also  Chabrier,  Vol  des  Insectes,  3Iem.  3Ius.  Hist.  Nat.  Paris,  vi-viii, 
1820-22. 


26  ON  MAGNITUDE  [ch. 

contained  in  the  equation  V  =  \^l,  a  result  which  happens  to  be 
identical  with  one  we  had  also  arrived  at  in  the  case  of  the  fish. 
In  the  bird's  case  it  has  a  deeper  significance  than  in  the  other; 
because  it  implies  here  not  merely  that  the  velocity  will  tend  to 
increase  in  a  certain  ratio  with  the  length,  but  that  it  must  do  so 
as  an  essential  and  primary  condition  of  the  bird's  remaining  aloft. 
It  is  accordingly  of  great  practical  importance  in  aeronautics,  for 
it  shews  how  a  provision  of  increasing  speed  must  accompany  every 
enlargement  of  our  aeroplanes.  If  a  given  machine  weighing,  say, 
500  lbs.  be  stable  at  40  miles  an  hour,  then  one  geometrically 
similar  which  weighs,  say,  a  couple  of  tons  must  have  its  speed 
determined  as  follows: 

W  -.wi-.L^:  P  ::  8  :  1. 

Therefore  L:l::2:l. 

But  72 :  ^2 ::  Z  :  I. 

Therefore  V  :  v.:  V2  :  1  =  1-414  :  1. 

That  is  to  say,  the  larger  machine  must  be  capable  of  a  speed 
equal  to  1-414  x  40,  or  about  56|  miles  per  hour. 

It  is  highly  probable,  as  Lanchester*  remarks,  that  Lilienthal 
met  his  untimely  death  not  so  much  from  any  intrinsic  fault  in 
the  design  or  construction  of  his  machine,  but  simply  because  his 
engine  fell  somewhat  short  of  the  power  required  to  give  the 
speed  which  was  necessary  for  stability.  An  arrow  is  a  very 
imperfectly  designed  aeroplane,  but  nevertheless  it  is  evidently 
capable,  to  a  certain  extent  and  at  a  high  velocity,  of  acquiring 
"stability"  and  hence  of  actual  "flight":  the  duration  and 
consequent  range  of  its  trajectory,  as  compared  with  a  bullet  of 
similar  initial  velocity,  being  correspondingly  benefited.  When 
we  return  to  our  birds,  and  again  compare  the  ostrich  with  the 
sparrow,  we  know  little  or  nothing  about  the  speed  in  flight  of 
the  latter,  but  that  of  the  swift  is  estimated  f  to  vary  from  a 
minimum  of  20  to  50  feet  or  more  per  second, — say  from  14  to 
35  miles  per  hour.  Let  us  take  the  same  lower  limit  as  not  far 
from  the  minimal  velocity  of  the  sparrow's  flight  also ;    and  it 

*  Aerial  Flight,  vol.  11  (Aerodonetics),  1908,  p.  150. 
f  By  Lanchester,  op.  cit.  p.  131. 


II]  THE   PRINCIPLE   OF   SIMILITUDE  27 

would  follow  that  the  ostrich,  of  25  times  the  sparrow's  linear 
dimensions,  would  be  compelled  to  fly  (if  it  flew  at  all)  with 
a  minimum  velocity  of  5  x  14,  or  70  miles  an  hour. 

The  same  principle  of  necessary  S'peed,  or  the  indispensable 
relation  between  the  dimensions  of  a  flying  object  and  the  minimum 
velocity  at  which  it  is  stable,  accounts  for  a  great  number  of 
observed  phenomena.  It  tells  us  why  the  larger  birds  have  a 
marked  difficulty  in  rising  from  the  ground,  that  is  to  say,  in 
acquiring  to  begin  with  the  horizontal  velocity  necessary  for  their 
support;  and  why  accordingly,  as  Mouillard*  and  others  have 
observed,  the  heavier  birds,  even  those  weighing  no  more  than 
a  pound  or  two,  can  be  effectively  "caged"  in  a  small  enclosure 
open  to  the  sky.  It  tells  us  why  very  small  birds,  especially 
those  as  small  as  humming-birds,  and  a  fortiori  the  still  smaller 
insects,  are  capable  of  "stationary  flight,"  a  very  sHght  and 
scarcely  perceptible  velocity  relatively  to  the  air  being  sufficient  for 
their  support  and  stability.  And  again,  since  it  is  in  all  cases 
velocity  relative  to  the  air  that  we  are  speaking  of,  we  comprehend 
the  reason  why  one  may  always  tell  which  way  the  wind  blows 
by  watching  the  direction  in  which  a  bird  starts  to  fly. 

It  is  not  improbable  that  the  ostrich  has  already  reached 
a  magnitude,  and  we  may  take  it  for  certain  that  the  moa  did 
so,  at  which  flight  by  muscular  action,  according  to  the  normal 
anatomy  of  a  bird,  has  become  physiologically  impossible.  The 
same  reasoning  applies  to  the  case  of  man.  It  would  be  very 
difficult,  and  probably  absolutely  impossible,  for  a  bird  to  fly 
were  it  the  bigness  of  a  man.  But  Borelli,  in  discussing  this 
question,  laid  even  greater  stress  on  the  obvious  fact  that  a  man's 
pectoral  muscles  are  so  immensely  less  in  proportion  than  those 
of  a  bird,  that  however  we  may  fit  ourselves  with  wings  we  can 
never  expect  to  move  them  by  any  power  of  our  own  relatively 
weaker  muscles ;  so  it  is  that  artificial  flight  only  became  possible 
when  an  engine  was  devised  whose  efficiency  was  extraordinarily 
great  in  comparison  with  its  weight  and  size. 

Had  Leonardo  da  Vinci  known  what  GaHleo  knew,  he  would 
not  have  spent  a  great  part  of  his  life  on  vain  efforts  to  make  to 
himself  wings.  Borelli  had  learned  the  lesson  thoroughly,  and 
*  Cf.  U empire  de  Vair  ;  oniitkologie  appliquee  a  Vaviation.  1881. 


28  ON  MAGNITUDE  [ch. 

in  one  of  his  chapters  he  deals  with  the  proposition,  "Est  im- 
possibile,  ut  homines  propriis  viribus  artificiose  volare  possint*." 

But  just  as  it  is  easier  to  swim  than  to  fly,  so  is  it  obvious 
that,  in  a  denser  atmosphere,  the  conditions  of  flight  would  be 
altered,  and  flight  facihtated.  We  know  that  in  the  carboniferous 
epoch  there  hved  giant  dragon-flies,  with  wings  of  a  span  far 
greater  than  nowadays  they  ever  attain;  and  the  small  bodies 
and  huge  extended  wings  of  the  fossil  pterodactyles  would  seem 
in  like  manner  to  be  quite  abnormal  according  to  our  present 
standards,  and  to  be  beyond  the  limits  of  mechanical  efficiency 
under  present  conditions.  But  as  Harle  suggests f,  following 
upon  a  suggestion  of  Arrhenius,  we  have  only  to  suppose  that  in 
carboniferous  and  Jurassic  days  the  terrestrial  atmosphere  was 
notably  denser  than  it  is  at  present,  by  reason,  for  instance,  of 
its  containing  a  much  larger  proportion  of  carbonic  acid,  and  we 
have  at  once  a  means  of  reconciling  the  apparent  mechanical 
discrepancy. 

Very  similar  problems,  involving  in  various  ways  the  principle 
of  dynamical  simihtude,  occur  all  through  the  physiology  of 
locomotion:  as,  for  instance,  when  we  see  that  a  cockchafer  can 
carry  a  plate,  many  times  his  own  weight,  upon  his  back,  or  that 
a  flea  can  jump  many  inches  high. 

Problems  of  this  latter  class  have  been  admirably  treated  both 
by  Gahleo  and  by  Borelli,  but  many  later  writers  have  remained 
ignorant  of  their  work.  Linnaeus,  for  instance,  remarked  that, 
if  an  elephant  were  as  strong  in  proportion  as  a  stag-beetle,  it 
would  be  able  to  pull  up  rocks  by  the  root,  and  to  level  mountains. 
And  Kirby  and  Spence  have  a  well-known  passage  directed  to 
shew  that  such  powers  as  have  been  conferred  upon  the  insect 
have  been  withheld  from  the  higher  animals,  for  the  reason  that 
had  these  latter  been  endued  therewith  they  would  have  "caused 
the  early  desolation  of  the  world  J." 

*  De  Motu  Animalium,  I,  prop,  cciv,  ed.  1685,  p.  243. 

f  Harle,  On  Atmospheric  Pressure  in  past  Geological  Ages,  Bull.  Geol.  Soc. 
Fr.  XI,  pp;  118-121;    or  Cosmos,  p.  30,  July  8,  1911. 

X  Introduction  to  Entomology,  1826,  ii,  p.  190.  K.  and  S.,  like  many  less  learned 
authors,  are  fond  of  popular  illustrations  of  the  "wonders  of  Nature,"  to  the  neglect 
of  dynamical  principles.  They  suggest,  for  instance,  that  if  the  white  ant  were 
as  big  as  a  man,  its  tunnels  would  be  "magnificent  cylinders  of  more  than  three 


II]  BORELLFS  LAW  29 

Such  problems  as  that  which  is  presented  by  the  flea's  jumping 
powers,  though  essentially  physiological  in  their  nature,  have  their 
interest  for  us  here:  because  a  steady,  progressive  diminution  of 
activity  with  increasing  size  would  tend  to  set  limits  to  the  possible 
growth  in  magnitude  of  an  animal  just  as  surely  as  those  factors 
which  tend  to  break  and  crush  the  living  fabric  under  its  own 
weight.  In  the  case  of  a  leap,  we  have  to  do  rather  with  a  sudden 
impulse  than  with  a  continued  strain,  and  this  impulse  should  be 
measured  in  terms  of  the  velocity  imparted.  The  velocity  is 
proportional  to  the  impulse  (x),  and  inversely  proportional  to  the 
mass  (M)  moved :  V  =  x/M.  But,  according  to  what  we  still  speak 
of  as  "  Borelli's  law,"  the  impulse  (i.e.  the  work  of  the  impulse)  is 
proportional  to  the  volume  of  the  muscle  by  which  it  is  produced  *, 
that  is  to  say  (in  similarly  constructed  animals)  to  the  mass  of  the 
whole  body ;  for  the  impulse  is  proportional  on  the  one  hand  to 
the  cross-section  of  the  muscle,  and  on  the  other  to  the  distance 
through  which  it  contracts.  It  follows  at  once  from  this  that  the 
velocity  is  constant,  whatever  be  the  size  of  the  animals :  in 
other  words,  that  all  animals,  provided  always  that  they  are 
similarly  fashioned,  with  their  various  levers  etc.,  in  like  proportion, 
ought  to  jump,  not  to  the  same  relative,  but  to  the  same  actual 
height f.  According  to  this,  then,  the  flea  is  not  a  better,  but 
rather  a  worse  jumper  than  a  horse  or  a  man.  As  a  matter  of 
fact,  Borelli  is  careful  to  point  out  that  in  the  act  of  leaping  the 
impulse  is  not  actually  instantaneous,  as  in  the  blow  of  a  hammer, 
but  takes  some  little  time,  during  which  the  levers  are  being 
extended  by  which  the  centre  of  gravity  of  the  animal  is  being 
propelled  forwards ;  and  this  interval  of  time  will  be  longer  in 
the  case  of  the  longer  levers  of  the  larger  animal.  To  some  extent, 
then,  this  principle  acts  as  a  corrective  to  the  more  general  one, 

hundred  feet  in  diameter";  and  that  if  a  certain  noisy  Brazilian  insect  were  as 
big  as  a  man,  its  voice  would  be  heard  aU  the  world  over:  "so  that  Stentor 
becomes  a  mute  when  compared  with  these  insects ! "  It  is  an  easy  consequence 
of  anthropomorphism,  and  hence  a  common  characteristic  of  fairy-tales,  to  neglect 
the  principle  of  dynamical,  while  dwelling  on  the  aspect  of  geometrical,  similarity. 

*  I.e.  the  available  energy  of  muscle,  in  ft. -lbs.  per  lb.  of  muscle,  is  the  same 
for  all  animals:  a  postulate  which  requires  considerable  qualification  when  we  are 
comparing  very  different  kinds  of  muscle,  such  as  the  insect's  and  the  mammal's. 

t  Prop,  clxxvii.  Animaha  minora  et  minus  ponderosa  majores  saltus  efficiunt 
respectu  sui  corporis,  si  caetera  fuerint  paria. 


30  ON  MAGNITUDE  [ch. 

and  tends  to  leave  a  certain  balance  of  advantage,  in  regard  to 
leaping  power,  on  the  side  of  the  larger  animal*. 

But  on  the  other  hand,  the  question  of  strength  of  materials 
comes  in  once  more,  and  the  factors  of  stress  and  strain  and 
bending  moment  make  it,  so  to  speak,  more  and  more  difficult 
for  nature  to  endow  the  larger  animal  with  the  length  of  lever 
with  which  she  has  provided  the  flea  or  the  grasshopper. 

To  Kirby  and  Spence  it  seemed  that  "  This  wonderful  strength 
of  insects  is  doubtless  the  result  of  something  pecuhar  in  the 
structure  and  arrangement  of  their  muscles,  and  principally  their 
extraordinary  power  of  contraction."'  This  hypothesis,  which  is 
so  easily  seen,  on  physical  grounds,  to  be  unnecessary,  has  been 
amply  disproved  in  a  series  of  excellent  papers  by  F.  Plateau  "f. 

A  somewhat  simple  problem  is  presented  to  us  by  the  act  of 
walking.  It  is  obvious  that  there  will  be  a  great  economy  of 
work,  if  the  leg  s'wing  at  its  normal  'pendnlum-rate;  and,  though 
this  rate  is  hard  to  calculate,  owing  to  the  shape  and  the  jointing 
of  the  limb,  we  may  easily  convince  ourselves,  by  counting  our 
steps,  that  the  leg  does  actually  swing,  or  tend  to  swing,  just  as 
a  pendulum  does,  at  a  certain  definite  rate  J.  When  we  walk 
quicker,  we  cause  the  leg-pendulum  to  describe  a  greater  arc,  but 
we  do  not  appreciably  cause  it  to  swing,  or  vibrate,  quicker,  until 
we  shorten  the  pendulum  and  begin  to  run.  Now  let  two  indi- 
viduals, A  and  B,  walk  in  a  similar  fashion,  that  is  to  say,  with 
a  similar  angle  of  swing.  The  arc  through  which  the  leg  swings, 
or  the  amplitude  of  each  step,  will  therefore  vary  as  the  length 
of  leg,  or  say  a?  ajh ;  but  the  time  of  swing  will  vary  as  the  square 

*  See  also  (int.  al.),  John  Bernoulli,  de  Motu  Musculorum,  Basil.,  1694; 
Chabrj',  Mecanisme  du  Saut,  J,  de  VAnat.  et  de  la  Physiol,  xix,  1883;  Sur  la 
longueur  des  membres  des  animaux  sauteurs,  ibid,  xxi,  p.  356,  1885;  Le  Hello, 
De  Taction  des  organes  locomoteurs,  etc.,  ibid,  xxix,  p.  65-93,  1893,  etc. 

t  Recherches  sur  la  force  absolue  des  muscles  des  Invertebres,  Bull.  Acad.  B. 
de  Belgique  (3),  vi,  \ti,  1883-84:  see  also  ibid.  (2),  xx,  1865,  xxii,  1866;  A^ut. 
Mag.  ]<:.  H.  x\ti,  p.  139,  1866,  xix,  p.  95,  1867.  The  subject  was  also  well  treated 
by  Straus-Diirckheim,  in  his  Considerations  generales  sur  Vanatomie  comparee  des 
animaux  articules,  1828. 

X  The  fact  that  the  limb  tends  to  swing  in  pendulum-time  was  first  observed 
by  the  brothers  Weber  [Mechanik  der  menschl.  Gehicerkzeuxje,  Gottingen,  1836). 
Some  later  writers  have  criticised  the  statement  (e.g.  Fischer,  Die  Kiuematik  des 
Beinschwingens  etc.,  Abh.  math.  phys.  Kl.  k.  Sachs.  Ges.  xxv-xxvm,  1899-1903), 
but  for  all  that,  with  proper  qualifications,  it  remains  substantially  true. 


II]  THE   PRINCIPLE   OF   SIMILITUDE  31 

root  of  the  peiidulum-length,  or  -^/ajy/'b.     Therefore  the  velocity, 

,  .  ,    .                    1  ,      amplitude        .,,     ,  , 

which  IS  measured  by          . ,  will  also  vary  as  the  square- 
time  ■  ^ 

roots  of  the  length  of  leg :   that  is  to  say,  the  average  velocities  of 

A  and  B  are  in  the  ratio  of  y'a  :  y/h. 

The  smaller  man,  or  smaller  animal,  is  so  far  at  a  disadvantage 
compared  with  the  larger  in  speed,  but  only  to  the  extent  of  the 
ratio  between  the  square  roots  of  their  linear  dimensions :  whereas, 
if  the  rate  of  movement  of  the  limb  were  identical,  irrespective 
of  the  size  of  the  animal, — if  the  limbs  of  the  mouse  for  instance 
swung  at  the  same  rate  as  those  of  the  horse, — then,  as  F.  Plateau 
said,  the  mouse  would  be  as  slow  or  slower  in  its  gait  than  the 
tortoise.  M.  Delisle*  observed  a  "minute  fly"  walk  three  inches 
in  half-a-second.  This  was  good  steady  walking.  When  we 
walk  five  miles  an  hour  we  go  about  88  inches  in  a  second,  or 
88/6  =  14-7  times  the  pace  of  M.  Dehsle's  fly.  We  should  walk 
at  just  about  the  fly's  pace  if  our  stature  were  1/(14-7)^,  or  1/216 
of  our  present  height, — say  72/216  inches,  or  one-third  of  an  inch 
high. 

But  the  leg  comprises  a  complicated  system  of  levers,  by  whose 
various  exercise  we  shall  obtain  very  different  results.  For 
instance,  by  being  careful  to  rise  upon  our  instep,  we  considerably 
increase  the  length  or  amplitude  of  our  stride,  and  very  considerably 
increase  our  speed  accordingly.  On  the  other  hand,  in  running, 
we  bend  and  so  shorten  the  leg,  in  order  to  accommodate  it  to 
a  quicker  rate  of  pendulum-swing "j".  In  short,  the  jointed  structure 
of  the  leg  permits  us  to  use  it  as  the  shortest  possible  pendulum 
when  it  is  swinging,  and  as  the  longest  possible  lever  when  it  is 
exerting  its  propulsive  force. 

Apart  from  such  modifications  as  that  described  in  the  last 
paragraph, — apart,  that  is  to  say,  from  differences  in  mechanical 
construction  or  in  the  manner  in  which  the  mechanism  is  used, — 
we  have  now  arrived  at  a  curiously  simple  and  uniform  result. 
For  in  all  the  three  forms  of  locomotion  which  we  have  attempted 

*  Quoted  in  Mr  John  Bishop's  interesting  article  in  Todd's  Cyclopaedia,  ni, 
p.  443. 

t  There  is  probably  also  another  factor  involved  here :  for  in  bending,  and  there- 
fore shortening,  the  leg  we  bring  its  centre  of  gravity  nearer  to  the  pivot,  that  is 
to  say,  to  the  joint,  and  so  the  muscle  tends  to  move  it  the  more  quickly. 


32  ON  MAGNITUDE  [ch. 

to  study,  alike  in  swimming,  in  flight  and  in  walking,  the  general 
result,  attained  under  very  different  conditions  and  arrived  at  by 
very  different  modes  of  reasoning,  is  in  every  case  that  the  velocity 
tends  to  vary  as  the  square  root  of  the  linear  dimensions  of  the 
organism. 

From  all  the  foregoing  discussion  we  learn  that,  as  Crookes 
once  upon  a  time  remarked*,  the  form  as  .well  as  the  actions  of  our 
bodies  are  entirely  conditioned  (save  for  certain  exceptions  in  the 
case  of  aquatic  animals,  nicely  balanced  with  the  density  of  the 
surrounding  medium)  by  the  strength  of  gravity  upon  this  globe. 
Were  the  force  of  gravity  to  be  doubled,  our  bipedal  form  would 
be  a  failure,  and  the  majority  of  terrestrial  animals  would  resemble 
short-legged  saurians,  or  else  serpents.  Birds  and  insects  would 
also  suffer,  though  there  would  be  some  compensation  for  them 
in  the  increased  density  of  the  air.  While  on  the  other  hand  if 
gravity  were  halved,  we  should  get  a  lighter,  more  graceful,  more 
active  type,  requiring  less  energy  and  less  heat,  less  heart,  less 
lungs,  less  blood. 

Throughout  the  whole  field  of  morphology  we  may  find 
examples  of  a  tendency  (referable  doubtless  in  each  case  to  some 
definite  physical  cause)  for  surface  to  keep  pace  with  volume, 
through  some  alteration  of  its  form.  The  development  of  "vilh" 
on  the  inner  surface  of  the  stomach  and  intestine  (which  enlarge 
its  surface  much  as  we  enlarge  the  effective  surface  of  a  bath- 
towel),  the  various  valvular  folds  of  the  intestinal  lining,  including 
the  remarkable  "spiral  fold"  of  the  shark's  gut,  the  convolutions 
of  the  brain,  whose  complexity  is  evidently  correlated  (in  part 
at  least)  with  the  magnitude  of  the  animal, — all  these  and  many 
more  are  cases  in  which  a  more  or  less  constant  ratio  tends  to  be 
maintained  between  mass  and  surface,  which  ratio  would  have 
been  more  and  mOre  departed  from  had  it  not  been  for  the 
alterations  of  surface-form  f. 

*  Proc.  Psychical  Soc.  xn,  pp.  338-355,  1897. 

f  For  various  calculations  of  the  increase  of  surface  due  to  histological  and 
anatomical  subdivision,  see  E.  Babak,  Ueber  die  Oberflachenentwickelung  bei 
Organismen,  Biol.  Centralbl.  xxs,  pp.  225-239,  257-267,  1910.  In  connection 
with  the  physical  theory  of  surface-energy,  Wolfgang  Ostwald  has  introduced  the 
conception  of  specific  surface,  that  is  to  say  the  ratio  of  surface  to  volume,  or  SjV. 
In  a  cube,  V=P,  and  S  =  61^;  therefore  SjV  —  Q/l.     Therefore  if  the  side  I  measure 


II]  OF   BULK   AND   SURFACE  33 

In  the  case  of  very  small  animals,  and  of  individual  cells,  the 
principle  becomes  especially  important,  in  consequence  of  the 
molecular  forces  whose  action  is  strictly  limited  to  the  superficial 
layer.  In  the  cases  just  mentioned,  action  is  facilitated  by  increase 
of  surface :  diffusion,  for  instance,  of  nutrient  liquids  or  respiratory 
gases  is  rendered  more  rapid  by  the  greater  area  of  surface ;  but 
there  are  other  cases  in  which  the  ratio  of  surface  to  mass  may 
make  an  essential  change  in  the  whole  condition  of  the  system. 
We  know,  for  instance,  that  iron  rusts  when  exposed  to  moist 
air,  but  that  it  rusts  ever  so  much  faster,  and  is  soon  eaten  away, 
if  the  iron  be  first  reduced  to  a  heap  of  small  filings ;  this  is  a 
mere  difference  of  degree.  But  the  spherical  surface  of  the  rain- 
drop and  the  spherical  surface  of  the  ocean  (though  both  happen 
to  be  aUke  in  mathematical  form)  are  two  totally  different  pheno- 
mena, the  one  due  to  surface-energy,  and  the  other  to  that  form 
of  mass-energy  which  we  ascribe  to  gravity.  The  contrast  is  still 
more  clearly  seen  in  the  case  of  waves :  for  the  httle  ripple,  whose 
form  and  manner  of  propagation  are  governed  by  surface-tension, 
is  found  to  travel  with  a  velocity  which  is  inversely  as  the  square 
root  of  its  length;  while  the  ordinary  big  waves,  controlled  by 
gravitation,  have  a  velocity  directly  proportional  to  the  square 
root  of  their  wave-length.  In  like  manner  we  shall  find  that  the 
form  of  all  small  organisms  is  largely  independent  of  gravity,  and 
largely  if  not  mainly  due  to  the  force  of  surface-tension :  either 
as  the  direct  result  of  the  continued  action  of  surface  tension  on 
the  semi-fluid  body,  or  else  as  the  result  of  its  action  at  a  prior 
stage  of  development,  in  bringing  about  a  form  which  subsequent 
chemical  changes  have  rendered  rigid  and  lasting.  In  either  case, 
we  shall  find  a  very  great  tendency  in  small  organisms  to  assume 
either  the  spherical  form  or  other  simple  forms  related  to  ordinary 
inanimate  surface-tension  phenomena ;  which  forms  do  not  recur 
in  the  external  morphology  of  large  animals,  or  if  they  in  part 
recur  it  is  for  other  reasons. 


6  cm.,  the  ratio  SjV  =  1,  and  such  a  cube  may  be  taken  as  our  standard,  or  unit 
of  specific  surface.  A  human  blood-corpuscle  has,  accordingly,  a  si^ecific  surface 
of  somewhere  about  14,000  or  15,000.  It  is  found  in  physical  chemistry  that 
surface  energy  becomes  an  important  factor  when  the  specific  surface  reaches  a 
value  of  10,000  or  thereby. 

T.  G.  3 


34  ON  MAGNITUDE  [ch. 

Now  this  is  a  very  important  matter,  and  is  a  notable  illustration 
of  that  principle  of  similitude  which  we  have  already  discussed 
in  regard  to  several  of  its  manifestations.  We  are  coming  easily 
to  a  conclusion  which  will  affect  the  whole  course  of  our  argument 
throughout  this  book,  namely  that  there  is  an  essential  difference 
in  kind  between  the  phenomena  of  form  in  the  larger  and  the 
smaller  organisms.  I  have  called  this  book  a  study  of  Growth 
and  Form,  because  in  the  most  familiar  illustrations  of  organic 
form,  as  in  our  own  bodies  for  example,  these  two  factors  are 
inseparably  associated,  and  because  we  are  here  justified  in  thinking 
of  form  as  the  direct  resultant  and  consequence  of  growth:  of 
growth,  whose  varying  rate  in  one  direction  or  another  has  pro- 
duced, by  its  gradual  and  unequal  increments,  the  successive 
stages  of  development  and  the  final  configuration  of  the  whole 
material  structure.  But  it  is  by  no  means  true  that  form  and 
growth  are  in  this  direct  and  simple  fashion  correlative  or  comple- 
mentary in  the  case  of  minute  portions  of  living  matter.  For  in 
the  smaller  organisms,  and  in  the  individual  cells  of  the  larger, 
we  have  reached  an  order  of  magnitude  in  which  the  intermolecular 
forces  strive  under  favourable  conditions  with,  and  at  length 
altogether  outweigh,  the  force  of  gravity,  and  also  those  other 
forces  leading  to  movements  of  convection  which  are  the  prevaihng 
factors  in  the  larger  material  aggregate. 

However  we  shall  require  to  deal  more  fully  with  this  matter 
in  our  discussion  of  the  rate  of  growth,  and  we  may  leave  it  mean- 
while, in  order  to  deal  with  other  matters  more  or  less  directly 
concerned  with  the  magnitude  of  the  cell. 

The  hving  cell  is  a  very  complex  field  of  energy,  and  of  energy 
of  many  kinds,  surface-energy  included.  Now  the  whole  surface- 
energy  of  the  cell  is  by  no  means  restricted  to  its  outer  surface; 
for  the  cell  is  a  very  heterogeneous  structure,  and  all  its  proto- 
plasmic alveoh  and  other  visible  (as  well  as  in\asible)  hetero- 
geneities make  up  a  great  system  of  internal  surfaces,  at  every 
part  of  which  one  "phase"  comes  in  contact  with  another  "phase," 
and  surface- energy  is  accordingly  manifested.  But  still,  the 
external  surface  is  a  definite  portion  of  the  system,  with  a  definite 
"phase"  of  its  own,  and  however  httle  we  may  know  of  the  distri- 
bution of  the  total  energy  of  the  system,  it  is  at  least  plain  that 


II]  THE   SIZE   OF  CELLS  35 

the  conditions  which  favour  equihbrium  will  be  greatly  altered  by 
the  changed  ratio  of  external  surface  to  mass  which  a  change  of 
magnitude,  unaccompanied  by  change  of  form,  produces  in  the  cell. 
Li  short,  however  it  may  be  brought  aboi^t,  the  phenomenon  of 
division  of  the  cell  will  be  precisely  what  is  required  to  keep 
approximately  constant  the  ratio  between  surface  and  mass,  and 
to  restore  the  balance  between  the  surface-energy  and  the  other 
energies  of  the  system.  When  a  germ-cell,  for  instance,  divides 
or  "'segments"  into  two,  it  does  not  increase  in  mass;  at  least  if 
there  be  some  shght  alleged  tendency  for  the  egg  to  increase  in 
mass  or  volume  during  segmentation,  it  is  very  slight  indeed, 
generally  imperceptible,  and  wholly  denied  by  some*.  The 
development  or  growth  of  the  egg  from  a  one-celled  stage  to 
stages  of  two  or  many  cells,  is  thus  a  somewhat  peculiar  kind 
of  growth;  it  is  growth  which  is  limited  to  increase  of  surface, 
unaccompanied  by  growth  in  volume  or  in  mass. 

In  the  case  of  a  soap-bubble,  by  the  way,  if  it  divide  into  two 
bubbles,  the  volume  is  actually  diminished f,  while  the  surface-area 
is  greatly  increased.  This  is  due  to  a  cause  which  we  shall  have 
to  study  later,  namely  to  the  increased  pressure  due  to  the  greater 
curvature  of  the  smaller  bubbles. 

An  immediate  and  remarkable  result  of  the  principles  just 
described  is  a  tendency  on  the  part  of  all  cells,  according  to  their 
kind,  to  vary  but  little  about  a  certain  mean  size,  and  to  have, 
in  fact,  certain  absolute  limitations  of  magnitude. 

Sachs  J  pointed  out,  in  1895,  that  there  is  a  tendency  for  each 
nucleus  to  be  only  able  to  gather  around  itself  a  certain  definite 
amount  of  protoplasm.  Driesch§,  a  little  later,  found  that,  by 
artificial  subdivision  of  the  egg,  it  was  possible  to  rear  dwarf 
sea-urchin  larvae,  one-half,  one-quarter,  or  even  one-eighth  of  their 

*  Though  the  entire  egg  is  not  increasing  in  mass,  this  is  not  to  say  that  its 
living  protoplasm  is  not  increasing  all  the  whUe  at  the  expense  of  tbe  reserve 
material. 

t  Of.  Tait,  Proc.  R.S.E.  v,  1866,  and  vi,  1868. 

X  Physiolog.  Notizen  (9),  p.  425,  1895.  Cf.  Strasbiirger,  Ueber  die  Wirkungs- 
sphare  der  Kerne  und  die  Zellgrosse,  Histolog.  Beitr.  (5),  pp.  95-129,  1893; 
J.  J.  Gerassimow,  Ueber  die  Grosse  des  Zellkernes,  Beih.  Bot.  Centralbl.  xvm, 
1905 ;  also  G.  Levi  and  T.  Terni,  Le  variazioni  dell'  indice  plasmatico-nucleare 
durante  1'  intercinesi.  Arch.  Ital.  di  Anat.  x,  p.  545,  1911. 

§  Arch.  f.  Entw.  Mech.  iv,  1898,  pp.  75,  247. 

3—2 


36  ON  MAGNITUDE  [ch. 

normal  size ;  and  that  these  dwarf  bodies  were  composed  of  only  a 
half,  a  quarter  or  an  eighth  of  the  normal  number  of  cells.  Similar 
observations  have  been  often  repeated  and  amply  confirmed.  For 
instance,  in  the  development  of  Crefidula  (a  httle  American 
"  shpper-hmpet,"  now  much  at  home  on  our  own  oyster-beds), 
Conkhn*  has  succeeded  in  rearing  dwarf  and  giant  individuals, 
of  which  the  latter  may  be  as  much  as  twenty-five  times  as  big 
as  the  former.  But  nevertheless,  the  individual  cells,  of  skin,  gut, 
liver,  muscle,  and  of  all  the  other  tissues,  are  just  the  same  size 
in  one  as  in  the  other, — in  dwarf  and  in  giant  f.  Driesch  has  laid 
particular  stress  upon  this  principle  of  a  "fixed  cell-size." 

We  get  an  excellent,  and  more  familiar  illustration  of  the  same 
principle  in  comparing  the  large  brain-cells  or  ganghon-cells,  both 
of  the  lower  and  of  the  higher  animalsf . 

In  Fig.  1  we  have  certain  identical  nerve-cells  taken  from 
various  mammals,  from  the  mouse  to  the  elephant,  all  represented 
on  the  same  scale  of  magnification ;  and  we  see  at  once  that  they 
are  all  of  much  the  same  order  of  magnitude.  The  nerve-cell  of 
the  elephant  is  about  twice  that  of  the  mouse  in  Unear  dimensions, 
and  therefore  about  eight  times  greater  in  volume,  or  mass.  But 
making  some  allow^ance  for  difference  of  shape,  the  Unear  dimen- 
sions of  the  elephant  are  to  those  of  the  mouse  in  a  ratio  certainly 
not  less  than  one  to  fifty;  from  which  it  would  follow  that  the 
bulk  of  the  larger  animal  is  something  like  125,000  times  that  of 
the  less.     And  it  also  follows,  the  size  of  the  nerve-cells  being 

*  Conklin,  E.  G.,  Cell-size  and  nuclear-size,  J.  Exp.  Zool.  xii.  pp.  1-98,  1912. 

t  Thus  the  fibres  of  the  crystalline  lens  are  of  the  same  size  in  large  and  small 
dogs ;  Eabl,  Z.  f.  w.  Z.  Lxvn,  1899.  Cf.  {i7it.  al.)  Pearson,  On  the  Size  of  the  Blood- 
corpuscles  in  Rana,  Biometrilca,  vi,  p.  403,  1909.  Dr  Thomas  Young  caught  sight 
of  the  phenomenon,  early  in  last  century:  "The  solid  particles  of  the  blood  do 
not  by  any  means  vary  in  magnitude  in  the  same  ratio  with  the  bulk  of  the  animal," 
Natural  Philosophy,  ed.  1845,  p.  466;  and  Leeuwenhoek  and  Stephen  Hales  were 
aware  of  it  a  hundred  years  before.  But  in  this  case,  though  the  blood-corpuscles 
show  no  relation  of  magnitude  to  the  size  of  the  animal,  they  do  seem  to  have  some 
relation  to  its  activity.  At  least  the  coipuscles  in  the  sluggish  Amphibia  are  much 
the  largest  known  to  us,  whQe  the  smallest  are  found  among  the  deer  and  other 
agile  and  speedy  mammals.  (Cf.  Gulliver,  P.Z.S.  1875,  p.  474,  etc.)  This  apparent 
correlation  may  have  its  bearing  on  modem  views  of  the  surface-condensation 
or  adsorption  of  oxygen  in  the  blood-corpuscles,  a  process  which  would  be  greatly 
facilitated  and  intensified  by  the  increase  of  surface  due  to  their  minuteness. 

%  Cf.  P.  Enriques,  La  forma  come  funzione  della  grandezza:  Ricerche  sui 
gangli  nervosi  degli  Invertebrati,  Arch.  f.  Entw.  Mech.  xxv,  p.  655,  1907-8. 


II]  THE   SIZE   OF   CELLS  37 

about  as  eight  to  one,  that,  in  corresponding  parts  of  the  nervous 
system  of  the  two  animals,  there  are  more  than  15,000  times  as 
many  individual  cells  in  one  as  in  the  other.  In  short  we  may 
(with  Enriques)  lay  it  down  as  a  general  law  that  among  animals, 
whether  large  or  small,  the  ganglion-cells  vary  in  size  within 
narrow  limits ;  and  that,  amidst  all  the  great  variety  of  structural 
type  of  ganglion  observed  in  different  classes  of  animals,  it  is 
always  found  that  the  smaller  species  have  simpler  ganglia  than 
the  larger,  that  is  to  say  ganglia  containing  a  smaller  number 
of  cellular  elements  *.  The  bearing  of  such  simple  facts  as  this 
upon  the  cell-theory  in  general  is  not  to  be  disregarded ;   and  the 


Horse 


Rabbit 


Fig.  1.     Motor  ganglion-cells,  from  the  cervical  spinal  cord. 
(From  Minot,  after  Irving  Hardesty.) 

warning  is  especially  clear  against  exaggerated  attempts  to 
correlate  physiological  processes  with  the  visible  mechanism  of 
associated  cells,  rather  than  with  the  system  of  energies,  or  the 
field  of  force,  which  is  associated  with  them.      For   the   life   of 

*  While  the  difference  in  cell-volume  is  vastly  less  than  that  between  the 
volumes,  and  very  much  less  also  than  that  between  the  surfaces,  of  the  respective 
animals,  yet  there  is  a  certain  diiJerenee ;  and  this  it  has  been  attempted  to  correlate 
with  the  need  for  each  cell  in  the  many-celled  ganglion  of  the  larger  animal  to 
possess  a  more  complex  "exchange-system"  of  branches,  for  intercommunication 
with  its  more  numerous  neighbours.  Another  explanation  is  based  on  the  fact 
that,  while  such  cells  as  continue  to  divide  throughout  life  tend  to  uniformity  of 
size  in  all  mammals,  those  which  do  not  do  so,  and  in  particular  the  ganglion  cells, 
continue  to  grow,  and  their  size  becomes,  therefore,  a  function  of  the  duration  of 
life.  Cf.  G.  Levi,  Studii  sulla  grandezza  delle  cellule,  Arch.  Iktl.  di  Anat.  e  di 
Emhryolog.  V,  p.  291,  1906. 


38  ON  MAGNITUDE  [ch. 

the  body  is  more  than  the  svm.  of  the  properties  of  the  cells  of 
which  it  is  composed:  as  Goethe  said,  "Das  Lebendige  ist  zwar 
in  Elemente  zerlegt,  aber  man  kann  es  aus  diesen  nicht  wieder 
zusammenstellen  und  beleben." 

Among  certain  lower  and  microscopic  organisms,  such  for 
instance  as  the  Rotifera,  we  are  still  more  palpably  struck  by  the 
small  number  of  cells  which  go  to  constitute  a  usually  complex 
organ,  such  as  kidney,  stomach,  ovary,  etc.  We  can  sometimes 
number  them  in  a  few  units,  in  place  of  the  thousands  that  make 
up  such  an  organ  in  larger,  if  not  always  higher,  animals.  These 
facts  constitute  one  among  many  arguments  which  combine  to 
teach  us  that,  however  important  and  advantageous  the  subdivision 
of  organisms  into  cells  may  be  from  the  constructional,  or  from 
the  dynamical  point  of  view,  the  phenomenon  has  less  essential 
importance  in  theoretical  biology  than  was  once,  and  is  often  still, 
assigned  to  it. 

Again,  just  as  Sachs  shewed  that  there  was  a  limit  to  the  amount 
of  cytoplasm  which  could  gather  round  a  single  nucleus,  so  Boveri 
has  demonstrated  that  the  nucleus  itself  has  definite  limitations 
of  size,  and  that,  in  cell-division  after  fertihsation,  each  new 
nucleus  has  the  same  size  as  its  parent-nucleus*. 

In  all  these  cases,  then,  there  are  reasons,  partly  no  doubt 
physiological,  but  in  very  large  part  purely  physical,  which  set 
hmits  to  the  normal  magnitude  of  the  organism  or  of  the  cell. 
But  as  we  have  already  discussed  the  existence  of  absolute  and 
definite  limitations,  of  a  physical  kind,  to  the  possible  increase  in 
magnitude  of  an  organism,  let  us  now  enquire  whether  there  be 
not  also  a  lower  limit,  below  which  the  very  existence  of  an 
organism  is  impossible,  or  at  least  where,  under  changed  conditions, 
its  very  nature  must  be  profoundly  modified. 

Among  the  smallest  of  known  organisms  we  have,  for  instance, 
Micromonas  mesnili,  Bonel,  a  flagellate  infusorian,  which  measures 
about  -M  fi,  or  •00034  mm.,  by  •00025  mm.;  smaller  even  than 
this  we  have  a  pathogenic  micrococcus  of  the  rabbit,  M.  pro- 
grediens,  Schrdter,  the  diameter  of  which  is  said  to  be  only  -00015 
mm.  or  -IS/x,  or  TS  x  10"^  cm., — about  equal  to  the  thickness  of 

*  Boveri,  Zellen-studien,  V.  Ueber  die  Ahhdngirjkeit  der  Kerncjrosse  und  Zellen- 
zahl  der  Seeigellarven  von  der  Chromosomenzakl  der  Ausgangszellert .     Jena,  1905. 


n] 


THE   LEAST   OF   ORGANISMS 


39 


the  thinnest  gold-leaf ;  and  as  small  if  not  smaller  still  are  a  few 
bacteria  and  their  spores.  But  here  we  have  reached,  or  all  but 
reached  the  utmost  limits  of  ordinary  microscopic  vision;  and 
there  remain  still  smaller  organisms,  the  so-called  "filter-passers," 
which  the  ultra-microscope  reveals,  but  which  are  mainly  brought 
within  our  ken  only  by  the  maladies,  such  as  hydrophobia,  foot- 
and-mouth  disease,  or  the  "mosaic"  disease  of  the  tobacco-plant, 
to  which  these  invisible  micro-organisms  give  rise*.     Accordingly, 


B 


B 


Fig.  2.  Relative  magnitudes  of:  A,  human  blood-corpuscle  (7-5^1  in  diameter); 
B,  Bacillus  anthracis  {4:  —  15 fx  x  I /j.)  ;  C.  various  Micrococci  (diam.  0-5  — l/n, 
rarely  2^i);    D,  Micromonas  progrediens,  Schroter  (diam.  0-15/x). 

since  it  is  only  by  the  diseases  which  they  occasion  that  these 
tiny  bodies  are  made  known  to  us,  we  might  be  tempted  to 
suppose  that  innumerable  other  invisible  organisms,  smaller  and 
yet  smaller,  exist  unseen  and  unrecognised  by  man. 

To  illustrate  some  of  these  small  magnitudes  I  have  adapted 
the  preceding  diagram  from  one  given  by  Zsigmondyf .    Upon  the 

*  Recent  important  researches  suggest  that  such  ultra-minute  '"filter-passers" 
are  the  true  cause  of  certain  acute  maladies  commonly  ascribed  to  the  presence 
of  much  larger  organisms;  cf.  Hort,  Lakin  and  Benians,  The  true  infective 
Agent  in  Cerebrospinal  Fever,  etc.,  J.  Roy.  Army  Med.  Corps,  Feb.  1916. 

■(•  Zur  Erkenntniss  cler  Kolloide,  1905,  p.  122;  where  there  wiU  be  found  an 
interesting  discussion  of  various  molecular  and  other  minute  magnitudes. 


40  ON  MAGNITUDE  [ch. 

same  scale  the  minute  iiltramicroscopic  particles  of  colloid  gold 
would  be  represented  by  the  finest  dots  which  we  could  make 
visible  to  the  naked  eye  upon  the  paper. 

A  bacillus  of  ordinary,  typical  size  is,  say,  1  /a  in  length.  The 
length  (or  height)  of  a  man  is  about  a  milhon  and  three-quarter 
times  as  great,  i.e.  1-75  metres,  or  1-75  x  10® /x;  and  the  mass  of 
the  man  is  in  the  neighbourhood  of  five  million,  million,  million 
(5  X  10^^)  times  greater  than  that  of  the  bacillus.  If  we  ask 
whether  there  may  not  exist  organisms  as  much  less  than  the 
bacillus  as  the  bacillus  is  less  than  the  dimensions  of  a  man,  it 
is  very  easy  to  see  that  this  is  quite  impossible,  for  we  are  rapidly 
approaching  a  point  where  the  question  of  molecular  dimensions, 
and  of  the  ultimate  divisibility  of  matter,  begins  to  call  for  our 
attention,  and  to  obtrude  itself  as  a  crucial  factor  in  the  case. 

Clerk  Maxwell  dealt  with  this  matter  in  his  article  ''Atom*,' 
and,  in  somewhat  greater  detail,  Errera  discusses  the  question  on 
the  following  lines f.  The  weight  of  a  hydrogen  molecule  is, 
according  to  the  physical  chemists,  somewhere  about  8*6  x  2  x  10-^^ 
milligrammes ;  and  that  of  any  other  element,  whose  molecular 
weight  is  M,  is  given  by  the  equation 

(M)  =  8-6  X  ilf  x  10-22. 
Accordingly,  the  weight  of  the  atom  of  sulphur  may  be  taken  as 
8-6  X  32  X  10-22  jj^gni.  =  275  x  IO-22  mgm. 

The  analysis  of  ordinary  bacteria  shews  them  to  consist  ±  of 
about  85  %  of  water,  and  15  %  of  solids;  while  the  solid  residue 
of  vegetable  protoplasm  contains  about  one  part  in  a  thousand 
of  sulphur.  We  may  assume,  therefore,  that  the  living  protoplasm 
contains  about 

Tooo   ^  Too  —  -L'J  A    iU 

parts  of  sulphur,  taking  the  total  weight  as  =  1. 

But  our  little  micrococcus,  of  0-15 /x  in  diameter,  would,  if  it 

were  spherical,  have  a  volume  of 

77 

7,  X  0-15^  u.,      =  18  X  10-^  cubic  microns: 

0 

*  Encyclopaedia  Britannica,  9th  edit.,  vol.  ill,  p.  42,  1875.  v 

t  Sur  la  limite  de  petitesse  des  organismes,  Bull.  Soc.  R.  des  Sc    med.  et  nat. 
de  Bruxelhs,  Jan.  1903 ;   Rec.  d'oeuvres  (Physiol,  generale),  p.  325. 
{  Cf.  A.  Fischer,  Vorlesungen  fiber  Bakterien,  1897,  p.  50. 


II]  THE  LEAST   OF   ORGANISMS  41 

and  therefore  (taking  its  density  as  equal  to  that  of  water),  a 
weight  of 

18  X  10-4  X  10-9  ==  18  X  10-13  mgm. 

But  of  this  total  weight,  the  sulphur  represents  only 

18  X  10-13  X  15  X  10-5  =  27  X  lO-i^  mgm. 

And  if  we  divide  this  by  the  weight  of  an  atom  of  sulphur,  we  have 

27  X  10-17  ^  275  X  10-22  _  io,000,  or  thereby. 

According  to  this  estimate,  then,  our  little  Micrococcus  frogrediens 
should  contain  only  about  10,000  atoms  of  sulphur,  an  element 
indispensable  to  its  protoplasmic  constitution ;  and  it  follows  that 
an  organism  of  one-tenth  the  diameter  of  our  micrococcus  would 
only  contain  10  sulphur-atoms,  and  therefore  only  ten  chemical 
"molecules"  or  units  of  protoplasm! 

It  may  be  open  to  doubt  whether  the  presence  of  sulphur  be 
really  essential  to  the  constitution  of  the  proteid  or  "  protoplasmic  " 
molecule ;  but  Errera  gives  us  yet  another  illustration  of  a 
similar  kind,  which  is  free  from  this  objection  or  dubiety.  The 
molecule  of  albumin,  as  is  generally  agreed,  can  scarcely  be  less 
than  a  thousand  times  the  size  of  that  of  such  an  element  as 
sulphur:  according  to  one  particular  determination*,  serum 
albumin  has  a  constitution  corresponding  to  a  molecular  weight 
of  10,166,  and  even  this  may  be  far  short  of  the  true  complexity 
of  a  typical  albuminoid  molecule.  The  weight  of  such  a  molecule  is 
8-6  X  10166  X  10-22  _  8-7  x  lO-i^  mgm. 

Now  the  bacteria  contain  about  14  %  of  albuminoids,  these 
constituting  by  far  the  greater  part  of  the  dry  residue;  and 
therefore  (from  equation  (5)),  the  weight  of  albumin  in  our  micro- 
coccus is  about 

^ij^  X  18  X  10-13  _  2-5  X  10-13  i^gii^_ 

If  we  divide  this  weight  by  that  which  we  have  arrived  at  as  the 
weight  of  an  albumin  molecule,  we  have 

2-5  X  10-13  ^  8-7  X  10-18  =  2-9  x  10*, 

in  other  w^ords,  our  micrococcus  apparently  contains  something 
less  than  30,000  molecules  of  albumin. 

*  F.  Hofmeister,  quoted  in  Cohnheim's  Chemie  der  Eiweisskorper,  1900,  p.  18. 


42  ON  MAGNITUDE  [ch. 

According  to  the  most  recent  estimates,  the  weight  of  the 
hydrogen  molecule  is  somewhat  less  than  that  on  which  Errera 
based  his  calculations,  namely  about  16  x  10"^^  mgms.  and 
according  to  this  value,  our  micrococcus  would  contain  just  about 
27,000  albumin  molecules.  In  other  words,  whichever  determina- 
tion we  accept,  we  see  that  an  organism  one-tenth  as  large  as  our 
micrococcus,  in  hnear  dimensions,  would  only  contain  some  thirty 
molecules  of  albumin ;  or,  in  other  words,  our  micrococcus  is  only 
about  thirty  times  as  large,  in  hnear  dimensions,  as  a  single  albumin 
molecule  *. 

We  must  doubtless  make  large  allowances  for  uncertainty  in 
the  assumptions  and  estimates  upon  which  these  calculations  are 
based ;  and  we  must  also  remember  that  the  data  with  which  the 
physicist  provides  us  in  regard  to  molecular  magnitudes  are,  to 
a  very  great  extent,  tnaximal  values,  above  which  the  molecular 
magnitude  (or  rather  the  sphere  of  the  molecule's  range  of  motion) 
is  not  Ukely  to  he :  but  below  which  there  is  a  greater  element  of 
uncertainty  as  to  its  possibly  greater  minuteness.  But  nevertheless^ 
when  we  shall  have  made  all  reasonable  allowances  for  uncertainty 
upon  the  physical  side,  it  will  still  be  clear  that  the  smallest  known 
bodies  which  are  described  as  organisms  draw  nigh  towards 
molecular  magnitudes,  and  we  must  recognise  that  the  subdivision 
of  the  organism  cannot  proceed  to  an  indefinite  extent,  and  in  all 
probability  cannot  go  very  much  further  than  it  appears  to  have 
done  in  these  already  discovered  forms.  For,  even  after  giving 
all  due  regard  to  the  complexity  of  our  unit  (that  is  to  say  the 
albumin-molecule),  with  all  the  increased  possibilities  of  inter- 
relation with  its  neighbours  which  this  complexity  impHes,  we 
cannot  but  see  that  physiologically,  and  comparatively  speaking, 
we  have  come  down  to  a  very  simple  thing. 

While  such  considerations  as  these,  based  on  the  chemical 
composition  of  the  organism,  teach  us  that  there  must  be  a  definite 
lower  hmit  to  its  magnitude,  other  considerations  of  a  purely 
physical  kind  lead  us  to  the  same  conclusion.  For  our  discussion 
of  the  principle  of  simihtude  has  already  taught  us  that,  long 
before    we    reach    these    almost    infinitesimal    magnitudes,    the 

*  McKendrick  arrived  at  a  still  lower  estimate,  of  about  1250  proteid  molecules 
in  the  minutest  organisms.     Brif.  Ass.  Rep.  1901,  p.  808. 


II]  THE   LEAST   OF   ORGANISMS  45 

diminishing  organism  will  have  greatly  changed  in  all  its  physical 
relations,  and  must  at  length  arrive  under  conditions  which  must 
surely  be  incompatible  with  anything  such  as  we  understand  by 
life,  at  least  in  its  full  and  ordinary  development  and  manifestation. 

We  are  told,  for  instance,  that  the  powerful  force  of  surface- 
tension,  or  capillarity,  begins  to  act  within  a  range  of  about 
1/500,000  of  an  inch,  or  say  0-05  ^u,.  A  soap-film,  or  a  film  of  oil 
upon  water,  may  be  attenuated  to  far  less  magnitudes  than  this ; 
the  black  spots  upon  a  soap-bubble  are  known,  by  various  con- 
cordant methods  of  measurement,  to  be  only  about  6  x  10-'^  cm.^ 
or  about  -006 /x  thick,  and  Lord  Rayleigh  and  M.  Devaux*  have 
obtained  films  of  oil  of  -002 /x,  or  even  -001  fi  in  thickness. 

But  while  it  is  possible  for  a  fluid  film  to  exist  in  these  almost 
molecular  dimensions,  it  is  certain  that,  long  before  we  reach 
them,  there  must  arise  new  conditions  of  which  we  have  Uttle 
knowledge  and  which  it  is  not  easy  even  to  imagine. 

It  would  seem  that,  in  an  organism  of  -1  /x  in  diameter,  or  even 
rather  more,  there  can  be  no  essential  distinction  between  the 
interior  and  the  surface  layers.  No  hollow  vesicle,  I  take  it,  can 
exist  of  these  dimensions,  or  at  least,  if  it  be  possible  for  it  to  do 
so,  the  contained  gas  or  fluid  must  be  under  pressures  of  a  formid- 
able kindf,  and  of  which  we  have  no  knowledge  or  experience. 
Nor,  I  imagine,  can  there  be  any  real  complexity,  or  heterogeneity, 
of  its  fluid  or  semi-fluid  contents ;  there  can  be  no  vacuoles  within 
such  a  cell,  nor  any  layers  defined  within  its  fluid  substance,  for 
something  of  the  nature  of  a  boundary-film  is  the  necessary 
condition  of  the  existence  of  such  layers.  Moreover,  the  whole 
organism,  provided  that  it  be  fluid  or  semi-fluid,  can  only  be 
spherical  in  form.  What,  then,  can  we  attribute,  in  the  way  of 
properties,  to  an  organism  of  a  size  as  small  as,  or  smaller  than, 
say  -05  [x  ?  It  must,  in  all  probability,  be  a  homogeneous,  structure- 
less sphere,  composed  of  a  very  small  number  of  albuminoid  or 
other  molecules.  Its  vital  properties  and  functions  must  be 
extraordinarily  limited ;  its  specific  outward  characters,  even  if  we 
could  see  it,  must  be  nil ;  and  its  specific  properties  must  be  httle 
more  than  those  of  an  ion-laden  corpuscle,  enabhng  it  to  perform 

*  Cf.  Perrin,  Les  Atomes,  1914,  p.  74. 

t   Cf.  Tait,  On  Compression  of  Air  in  small  'Bubbles,  Proc.  B.  S.  E.  V,  1865. 


44  ON  MAGNITUDE  [ch. 

this  or  that  chemical  reaction,  or  to  produce  this  or  that  patho- 
genic effect.  Even  among  inorganic,  non-Hving  bodies,  there 
must  be  a  certain  grade  of  minuteness  at  which  the  ordinary 
properties  become  modified.  For  instance,  while  under  ordinary 
circumstances  crystaUisation  starts  in  a  solution  about  a  minute 
solid  fragment  or  crystal  of  the  salt,  Ostwald  has  shewn  that  we 
may  have  particles  so  minute  that  they  fail  to  serve  as  a  nucleus 
for  crystallisation, — which  is  as  much  as  to  say  that  they  are  too 
minute  to  have  the  form  and  properties  of  a  " crystal"  ;  and  again, 
in  his  thin  oil-films.  Lord  Rayleigh  has  noted  the  striking  change 
of  physical  properties  which  ensues  when  the  film  becomes 
attenuated  to  something  less  than  one  close-packed  layer  of 
molecules*. 

Thus,  as  Clerk  Maxwell  put  it,  "molecular  science  sets  us  face 
to  face  with  physiological  theories.  It  forbids  the  physiologist 
from  imagining  that  structural  details  of  infinitely  small  dimensions 
[such  as  Leibniz  assumed,  one  within  another,  ad  ivfiiiitiim] 
can  furnish  an  explanation  of  the  infinite  variety  which  exists  in 
the  properties  and  functions  of  the  most  minute  organisms." 
And  for  this  reason  he  reprobates,  with  not  undue  severity,  those 
advocates  of  pangenesis  and  similar  theories  of  heredity,  who 
would  place  "a  whole  world  of  wonders  within  a  body  so  small 
and  so  devoid  of  visible  structure  as  a  germ."  But  indeed  it 
scarcely  needed  Maxwell's  criticism  to  shew  forth  the  immense 
physical  difficulties  of  Darwin's  theory  of  Pangenesis :  which, 
after  all,  is  as  old  as  Democritus,  and  is  no  other  than  that 
Promethean  particulam  tmdique  desectani  of  which  we  have  read, 
and  at  which  we  have  smiled,  in  our  Horace. 

There  are  many  other  ways  in  which,  when  we  "make  a  long 
excursion  into  space,"  we  find  our  ordinary  rules  of  physical 
behaviour  entirely  upset.  A  very  familiar  case,  analysed  by 
Stokes,  is  that  the  viscosity  of  the  surrounding  medium  has  a 
relatively  powerful  effect  upon  bodies  below  a  certain  size. 
A  droplet  of  water,  a  thousandth  of  an  inch  (25  /x)  in  diameter, 
cannot  fall  in  still  air  quicker  than  about  an  inch  and  a  half  per 
second;  and  as  its  size  decreases,  its  resistance  varies  as  the 
diameter,  and  not  (as  with  larger  bodies)  as  the  surface  of  the 

,      *  Phil.  Mag.  XLvni,  1*899 ;    Collected  Papers,  iv.  p.  430. 


II]  OF  MOLECULAK  MAGNITUDES  45 

drop.  Thus  a  drop  one-tenth  of  that  size  (2-5  ^u.),  the  size, 
apparently,  of  the  drops  of  water  in  a  Ught  cloud,  will  fall  a 
hundred  times  slower,  or  say  an  inch  a  minute;  and  one  again 
a  tenth  of  this  diameter  (say  -25  jj,,  or  about  twice  as  big,  in  linear 
dimensions,  as  our  micrococcus),  will  scarcely  fall  an  inch  in  two 
hours.  By  reason  of  this  principle,  not  only  do  the  smaller 
bacteria  fall  very  slowly  through  the  air,  but  all  minute  bodies 
meet  with  great  proportionate  resistance  to  their  movements  in 
a  fluid.  Even  such  comparatively  large  organisms  as  the  diatoms 
and  the  foraminifera,  laden  though  they  are  with  a  heavy  shell 
of  flint  or  lime,  seem  to  be  poised  in  the  water  of  the  ocean,  and 
fall  in  it  with  exceeding  slowness. 

The  Brownian  movement  has  also  to  be  reckoned  with, — that 
remarkable  phenomenon  studied  nearly  a  century  ago  (1827)  by 
Kobert  Brown,  facile  frincefs  botanicorum.  It  is  one  more  of  those 
fundamental  physical  phenomena  which  the  biologists  have  con- 
tributed, or  helped  to  contribute,  to  the  science  of  physics. 

The  quivering  motion,  accompanied  by  rotation,  and  even  by 
translation,  manifested  by  the  fine  granular  particles  issuing  from 
a  crushed  pollen-grain,  and  which  Robert  Brown  proved  to  have 
no  vital  significance  but  to  be  manifested  also  by  all  minute 
particles  whatsoever,  organic  and  inorganic,  was  for  many  years 
unexplained.  Nearly  fifty  years  after  Brown  wrote,  it  was  said 
to  be  "due,  either  directly  to  some  calorical  changes  continually 
taking  place  in  the  fluid,  or  to  some  obscure  chemical  action 
between  the  soUd  particles  and  the  fluid  w^hich  is  indirectly 
promoted  by  heat*."  Very  shortly  after  these  last  words  were 
written,  it  was  ascribed  by  Wiener  to  molecular  action,  and  we 
now  know  that  it  is  indeed  due  to  the  impact  or  bombardment  of 
molecules  upon  a  body  so  small  that  these  impacts  do  not  for 
the  moment,  as  it  were,  "average  out"  to  approximate  equality 
on  all  sides.  The  movement  becomes  manifest  with  particles  of 
somewhere  about  20  jj.  in  diameter,  it  is  admirably  displayed  by 
particles  of  about  12/^  in  diameter,  and  becomes  more  marked 
the  smaller  the  particles  are.  The  bombardment  causes  our 
particles  to  behave  just  Hke  molecules  of  uncommon  size,  and  this 

*  Carpenter,  The  Microscope,  erlit.  1862,  p.  185. 


46  ON  MAGNITUDE  [ch. 

behaviour  is  manifested  in  several  ways*.  Firstly,  we  have  the 
quivering  movement  of  the  particles;  secondly,  their  movement 
backwards  and  forwards,  in  short,  straight,  disjointed  paths ; 
thirdly,  the  particles  rotate,  and  do  so  the  more  rapidly  the  smaller 
they  are,  and  by  theory,  confirmed  by  observation,  it  is  found 
that  particles  of  1  /a  in  diameter  rotate  on  an  average  through 
100°  per  second,  while  particles  of  13 /a  in  diameter  turn  through 
only  14°  per  minute.  Lastly,  the  very  curious  result  appears,  that 
in  a  layer  of  fluid  the  particles  are  not  equally  distributed,  nor  do 
they  all  ever  fall,  under  the  influence  of  gravity,  to  the  bottom. 
But  just  as  the  molecules  of  the  atmosphere  are  so  distributed, 
under  the  influence  of  gravity,  that  the  density  (and  therefore  the 
number  of  molecules  per  unit  volume)  falls  off  in  geometrical 
progression  as  we  ascend  to  higher  and  higher  layers,  so  is  it  with 
our  particles,  even  within  the  narrow  limits  of  the  little  portion 
of  fluid  under  our  microscope.  It  is  only  in  regard  to  particles 
of  the  simplest  form  that  these  phenomena  have  been  theoretically 
investigated!,  and  we  may  take  it  as  certain  that  more  complex 
particles,  such  as  the  twisted  body  of  a  Spirillum,  would  show 
other  and  still  more  compUcated  manifestations.  It  is  at  least 
clear  that,  just  as  the  early  microscopists  in  the  days  before  Robert 
Brown  never  doubted  but  that  these  phenomena  were  purely 
vital,  so  we  also  may  still  be  apt  to  confuse,  in  certain  cases,  the 
one  phenomenon  with  the  other.  We  cannot,  indeed,  without  the 
most  careful  scrutiny,  decide  whether  the  movements  of  our 
minutest  organisms  are  intrinsically  "vital"  (in  the  sense  of  being 
beyond  a  physical  mechanism,  or  working  model)  or  not.  For  ex- 
ample, Schaudinn  has  suggested  that  the  undulating  movements  of 
Sfirochaete  pallida  must  be  due  to  the  presence  of  a  minute,  unseen, 
"undulating  membrane";  and  Doflein  says  of  the  same  species 
that  "sie  verharrt  oft  mit  eigenthiimlich  zitternden  Bewegungen 
zu  einem  Orte."     Both  movements,  the  trembhng  or  quivering 

*  The  modern  literature  on  the  Brownian  Movement  is  very  large,  owing  to  the 
value  which  the  phenomenon  is  shewn  to  have  in  determining  the  size  of  the  atom. 
For  a  fuUer,  but  still  elementary  account,  see  J.  Cox,  Beyond  the  Atom,  1913, 
pp.  118-128;  and  see,  further,  Perrin,  Les  Atomes,  pp.  119-189. 

•j-  Cf.  R.  Gans,  Wie  fallen  Stabe  mid  Scheiben  m  euier  reibenden  Fliissigkeit  ? 
Munchener  Bericht,  1911,  p.  191;  K.  Przibram,  Ueber  die  Brown'sche  Bewegung 
nicht  kugelformiger  Teilchen,  Wiener  Ber.  1912,  p.  2339. 


II],        THE  BROWNIAN  MOVEMENT         47 

movement  described  by  Doflein,  and  the  undulating  or  rotating 
movement  described  by  Schaudinn,  are  just  such  as  may  be  easily 
and  naturally  interpreted  as  part  and  parcel  of  the  Brownian 
phenomenon. 

While  the  Brownian  movement  may  thus  simulate  in  a  deceptive 
way  the  active  movements  of  an  organism,  the  reverse  statement 
also  to  a  certain  extent  holds  good.  One  sometimes  lies  awake  of 
a  summer's  morning  watching  the  flies  as  they  dance  under  the 
ceiling.  It  is  a  very  remarkable  dance.  The  dancers  do  not 
whirl  or  gyrate,  either  in  company  or  alone ;  but  they  advance 
and  retire;  they  seem  to  jostle  and  rebound;  between  the  rebounds 
they  dart  hither  or  thither  in  short  straight  snatches  of  hurried 
flight;  and  turn  again  sharply  in  a  new  rebound  at  the  end  of  each 
little  rush.  Their  motions  are  wholly  "erratic,"  independent  of 
one  another,  and  devoid  of  common  purpose.  This  is  nothing  else 
than  a  vastly  magnified  picture,  or  simulacrum,  of  the  Brownian 
movement;  the  parallel  between  the  two  cases  lies  in  their 
complete  irregularity,  but  this  in  itself  implies  a  close  resemblance. 
One  might  see  the  same  thing  in  a  crowded  market-place,  always 
provided  that  the  bustling  crowd  had  no  business  whatsoever. 
In  like  manner  Lucretius,  and  Epicurus  before  him,  watched  the 
dust-motes  quivering  in  the  beam,  and  saw  in  them  a  mimic 
representation,  rei  simulacrmn  ef  imago,  of  the  eternal  motions  of 
the  atoms.  Again  the  same  phenomenon  may  be  witnessed  under 
the  microscope,  in  a  drop  of  water  swarming  with  Paramoecia  or 
suchlike  Infusoria  ;  and  here  the  analogy  has  been  put  to  a  numerical 
test.  Following  with  a  pencil  the  track  of  each  little  swimmer, 
and  dotting  its  place  every  few  seconds  (to  the  beat  of  a  metronome), 
Karl  Przibram  found  that  the  mean  successive  distances  from  a 
common  base-line  obeyed  with  great  exactitude  the  "Einstein 
formula,"  that  is  to  say  the  particular  form  of  the  "law  of  chance" 
which  is  applicable  to  the  case  of  the  Brownian  movement*.  The 
phenomenon  is  (of  course)  merely  analogous,  and  by  no  means 
identical  with  the  Brownian  movement;  for  the  range  of  motion 
of  the  little  active  organisms,  whether  they  be  gnats  or  infusoria, 
is  vastly  greater  than  that  of  the  minute  particles  which  are 

*  Ueber  die  ungeordnete  Bewegung  niederer  Thiere,  Pfliiger^s  Archiv,  CLin, 
p.  401,  1913. 


48  ON  MAGNITUDE  [ch. 

passive  under  bombardment;  but  nevertheless  Przibram  is 
inclined  to  think  that  even  his  comparatively  large  infusoria  are 
small  enough  for  the  molecular  bombardment  to  be  a  stimulus, 
though  not  the  actual  cause,  of  their  irregular  and  interrupted 
movements. 

There  is  yet  another  very  remarkable  phenomenon  which  may 
come  into  play  in  the  case  of  the  minutest  of  organisms ;  and  this 
is  their  relation  to  the  rays  of  light,  as  Arrhenius  has  told  us. 
On  the  waves  of  a  beam  of  light,  a  very  minute  particle  {in 
vacuo)  should  be  actually  caught  up,  and  carried  along  with 
an  immense  velocity;  and  this  "radiant  pressure"  exercises 
its  most  powerful  influence  on  bodies  which  (if  they  be  of 
spherical  form)  are  just  about  -00016  mm.,  or  -16 /x  in  diameter. 
This  is  just  about  the  size,  as  we  have  seen,  of  some  of 
our  smallest  known  protozoa  and  bacteria,  while  we  have 
some  reason  to  believe  that  others  yet  unseen,  and  perhaps 
the  spores  of  many,  are  smaller  still.  Now  we  have  seen  that 
such  minute  particles  fall  with  extreme  slowness  in  air,  even  at 
ordinary  atmospheric  pressures:  our  organism  measuring  '16 /j, 
would  fall  but  83  metres  in  a  year,  which  is  as  much  as  to  say 
that  its  weight  offers  practically  no  impediment  to  its  transference, 
by  the  slightest  current,  to  the  very  highest  regions  of  the  atmo- 
sphere. Beyond  the  atmosphere,  however,  it  cannot  go,  until 
some  new  force  enable  it  to  resist  the  attraction  of  terrestrial 
gravity,  which  the  viscosity  of  an  atmosphere  is  no  longer  at 
hand  to  oppose.  But  it  is  conceivable  that  our  particle  ?nay  ga 
yet  farther,  and  actually  break  loose  from  the  bonds  of  earth. 
For  in  the  upper  regions  of  the  atmosphere,  say  fifty  miles  high, 
it  will  come  in  contact  with  the  rays  and  flashes  of  the  Northern 
Lights,  which  consist  (as  Arrhenius  maintains)  of  a  fine  dust,  or 
cloud  of  vapour-drops,  laden  with  a  charge  of  negative  electricity, 
and  projected  outwards  from  the  sun.  As  soon  as  our  particle 
acquires  a  charge  of  negative  electricity  it  will  begin  to  be  repelled 
by  the  similarly  laden  auroral  particles,  and  the  amount  of  charge 
necessary  to  enable  a  particle  of  given  size  (such  as  our  little 
monad  of  -16  /x)  to  resist  the  attraction  of  gravity  may  be  calculated, 
and  is  found  to  be  such  as  the  actual  conditions  can  easily  supply. 
Finally,  when  once  set  free  from  the  entanglement  of  the  earth's 


II]  THE   PRESSURE   OF  LIGHT  49 

atmosphere,  the  particle  may  be  propelled  by  the  "radiant 
pressure "  of  light,  with  a  velocity  which  will  carry  it. — like 
Uriel  gliding  on  a  sunbeam, — as  far  as  the  orbit  of  Mars  in 
twenty  days,  of  Jupiter  in  eighty  days,  and  as  far  as  the  nearest 
fixed  star  in  three  thousand,  years !  This,  and  much  more,  is 
Arrhenius's  contribution  towards  the  acceptance  of  Lord  Kelvin's 
hypothesis  that  life  may  be,  and  may  have  been,  disseminated 
across  the  bounds  of  space,  throughout  the  solar  system  and  the 
whole  universe ! 

It  may  well  be  that  we  need  attach  no  great  practical  importance 
to  this  bold  conception ;  for  even  though  stellar  space  be  shewn  to 
be  mare  liberum  to  minute  material  travellers,  we  may  be  sure  that 
those  which  reach  a  stellar  or  even  a  planetary  bourne  are  infinitely, 
or  all  but  infinitely,  few.  But  whether  or  no,  the  remote  possibilities 
of  the  case  serve  to  illustrate  in  a  very  vivid  way  the  profound 
differences  of  physical  property  and  potentiahty  which  are 
associated  in  the  scale  of  magnitude  with  simple  differences  of 
degree. 


CHAPTER   III 

THE   RATE   OF   GROWTH 

When  we  study  magnitude  by  itself,  apart,  that  is  to  say, 
from  the  gradual  changes  to  which  it  may  be  subject,  we  are 
deaUng  Avith  a  something  which  may  be  adequately  represented 
by  a  number,  or  by  means  of  a  hne  of  definite  length ;  it  is  what 
mathematicians  call  a  scalar  phenomenon.  When  we  introduce 
the  conception  of  change  of  magnitude,  of  magnitude  which  varies 
as  we  pass  from  one  direction  to  another  in  space,  or  from  one 
instant  to  another  in  time,  our  phenomenon  becomes  capable  of 
representation  by  means  of  a  line  of  which  we  define  both  the 
length  and  the  direction ;  it  is  (in  this  particular  aspect)  what  is 
called  a  vector  phenomenon. 

When  we  deal  with  magnitude  in  relation  to  the  dimensions 
of  space,  the  vector  diagram  which  we  draw  plots  magnitude  in 
one  direction  against  magnitude  in  another, — length  against 
height,  for  instance,  or  against  breadth ;  and  the  result  is  simply 
what  we  call  a  picture  or  drawing  of  an  object,  or  (more  correctly) 
a  "plane  projection"  of  the  object.  In  other  words,  what  we 
call  Form  is  a  ratio  of  magnitudes,  referred  to  direction  in  space. 

When  in  deahng  with  magnitude  we  refer  its  variations  to 
successive  intervals  of  time  (or  when,  as  it  is  said,  we  equate  it 
with  time),  we  are  then  dealing  with  the  phenomenon  of  groivth ; 
and  it  is  evident,  therefore,  that  this  term  growth  has  wide 
meanings.  For  growth  may  obviously  be  positive  or  negative ; 
that  is  to  say,  a  thing  may  grow  larger  or  smaller,  greater  or  less ; 
and  by  extension  of  the  primitive  concrete  signification  of  the 
word,  we  easily  and  legitimately  apply  it  to  non-material  things, 
such  as  temperature,  and  say,  for  instance,  that  a  body  "grows" 
hot  or  cold.  When  in  a  two-dimensional  diagram,  we  represent 
a  magnitude  (for  instance  length)  in  relation  to  time  (or  "plot" 


CH.  Ill]  CONCERNING   DIMENSIONS  51 

length  against  time,  as  the  phrase  is),  we  get  that  kind  of  vector 
diagram  which  is  commonly  known  as  a  "curve  of  growth."  We 
perceive,  accordingly,  that  the  phenomenon  which  we  are  now 

studying  is  a  velocity  (whose  "  dimensions"  are  ^t or  ^  J  ;  and 

this  phenomenon  we  shall  speak  of,  simply,  as  a  rate  of  growth. 

In  various  conventional  ways  we  can  convert  a  two-dimensional 
into  a  three-dimensional  diagram.  We  do  so,  for  example,  by 
means  of  the  geometrical  method  of  "perspective"  when  we 
represent  upon  a  sheet  of  paper  the  length,  breadth  and  depth  of 
an  object  in  three-dimensional  space ;  but  we  do  it  more  simply, 
as  a  rule,  by  means  of  "contour-lines,"  and  always  when  time  is 
one  of  the  dimensions  to  be  represented.  If  we  superimpose  upon 
one  another  (or  even  set  side  by  side)  pictures,  or  plane  projections, 
of  an  organism,  drawn  at  successive  intervals  of  time,  we  have 
such  a  three-dimensional  diagram,  which  is  a  partial  representation 
(hmited  to  two  dimensions  of  space)  of  the  organism's  gradual 
change  of  form,  or  course  of  development;  and  in  such  a  case 
our  contour-lines  may,  for  the  purposes  of  the  embryologist,  be 
separated  by  intervals  representing  a  few  hours  or  days,  or,  for 
the  purposes  of  the  palaeontologist,  by  interspaces  of  unnumbered 
and  innumerable  years*. 

Such  a  diagram  represents  in  two  of  its  three  dimensions  form, 
and  in  two,  or  three,  of  its  dimensions  growth ;  and  so  we  see  how 
intimately  the  two  conceptions  are  correlated  or  iriter-related  to 
one  another.  In  short,  it  is  obvious  that  the  form  of  an  animal 
is  determined  by  its  specific  rate  of  growth  in  various  directions ; 
accordingly,  the  phenomenon  of  rate  of  growth  deserves  to  be 
studied  as  a  necessary  preliminary  to  the  theoretical  study  of 
form,  and,  mathematically  speaking,  organic  foym  itself  appears 
to  us  as  a  function  of  titnei. 

*  Sometimes  we  find  one  and  the  same  diagram  suffice,  whether  the  intervals 
of  time  be  great  or  small;  and  we  then  invoke  "Wolff's  Law,"  and  assert  that 
the  life-history  of  the  individual  repeats,  or  recapitulates,  the  history  of  the  race. 

f  Our  subject  is  one  of  Bacon's  "Instances  of  the  Course,"  or  studies  wherein 
we  "measure  Nature  by  periods  of  Time."  In  Bacon's  Catalogue  of  Particular 
Histories,  one  of  the  odd  hundred  histories  or  investigations  which  he  foreshadowed 
is  precisely  that  which  we  are  engaged  on,  viz.  a  "History  of  the  Growth  and  Increase 
of  the  Body,  in  the  whole  and  in  its  parts." 

4—2 


52  •  THE  RATE   OF   GROWTH  [ch. 

At  the  same  time,  we  need  only  consider  this  part  of  our 
subject  somewhat  briefly.  Though  it  has  an  essential  bearing  on 
the  problems  of  morphology,  it  is  in  greater  degree  involved  with 
physiological  problems ;  and  furthermore,  the  statistical  or 
numerical  aspect  of  the  question  is  peculiarly  adapted  for  the 
mathematical  study  of  variation  and  correlation.  On  these 
important  subjects  we  shall  scarcely  touch ;  for  our  main  purpose 
will  be  sufficiently  served  if  we  consider  the  characteristics  of  a 
rate  of  growth  in  a  few  illustrative  cases,  and  recognise  that  this 
rate  of  growth  is  a  very  important  specific  property,  with  its  own 
characteristic  value  in  this  organism  or  that,  in  this  or  that  part 
of  each  organism,  and  in  this  or  that  phase  of  its  existence. 

The  statement  which  we  have  just  made  that  "the  form  of  an 
organism  is  determined  by  its  rate  of  growth  in  various  directions," 
is  one  which  calls  (as  we  have  partly  seen  in  the  foregoing  chapter) 
for  further  explanation  and  for  some  measure  of  qualification. 
Among  organic  forms  we  shall  have  frequent  occasion  to  see  that 
form  is  in  many  cases  due  to  the  immediate  or  direct  action  of 
certain  molecular  forces,  of  which  surface-tension  is  that  which  plays 
the  greatest  part.  Now  when  surface-tension  (for  instance)  causes 
a  minute  semi-fluid  organism  to  assume  a  spherical  form,  or  gives 
the  form  of  a  catenary  or  an  elastic  curve  to  a  film  of  protoplasm 
in  contact  with  some  sohd  skeletal  rod,  or  when  it  acts  in  various 
other  ways  which  are  productive  of  definite  contours,  this  is  a  pro- 
cess of  conformation  that,  both  in  appearance  and  reahty,  is  very 
difierent  from  the  process  by  which  an  ordinary  plant  or  animal 
grows  into  its  specific  form.  In  both  cases,  change  of  form  is 
brought  about  by  the  movement  of  portions  of  matter,  and  in 
both  cases  it  is  ultimately  due  to  the  action  of  molecular  forces ; 
but  in  the  one  case  the  movements  of  the  particles  of  matter  lie 
for  the  most  part  within  molecular  range,  while  in  the  other  we. 
have  to  deal  chiefly  with  the  transference  of  portions  of  matter 
into  the  system  from  without,  and  from  one  widely  distant  part 
of  the  organism  to  another.  It  is  to  this  latter  class  of  phenomena 
that  we  usually  restrict  the  term  growth;  and  it  is  in  regard  to 
them  that  we  are  in  a  position  to  study  the  rate  of  action  in 
different  directions,  and  to  see  that  it  is  merely  on  a  difference 
of  velocities  that  the  modification  of  form  essentially  depends. 


Ill]  OF  MOLAR   AND   MOLECULAR  FORCES  53 

The  difference  between  the  two  classes  ^i  phenomena  is  somewhat 
akin  to  the  difference  between  the  forces  which  determine  the 
form  of  a  rain-drop  and  those  which,  by  the  flowing  of  the  waters 
and  the  sculpturing  of  the  sohd  earth,  have  brought  about  the 
complex  configuration  of  a  river ;  molecular  forces  are  paramount 
in  the  conformation  of  the  one,  and  molar  forces  are  dominant 
in  the  other. 

At  the  same  time  it  is  perfectly  true  that  all  changes  of  form, 
inasmuch  as  they  necessarily  involve  changes  of  actual  and  relative 
magnitude,  may,  in  a  sense,  be  properly  looked  upon  as  phenomena 
of  growth ;  and  it  is  also  true,  since  the  movement  of  matter  must 
always  involve  an  element  of  time*,  that  in  all  cases  the  rate  of 
growth  is  a  phenomenon  to  be  considered.  Even  though  the 
molecular  forces  which  play  their  part  in  modifying  the  form  of 
an  organism  exert  an  action  which  is,  theoretically,  all  but 
instantaneous,  that  action  is  apt  to  be  dragged  out  to  an  appreciable 
interval  of  time  by  reason  of  viscosity  or  some  other  form  of 
resistance  in  the  material.  From  the  physical  or  physiological 
point  of  view  the  rate  of  action  even  in  such  cases  may  be  well 
worth  studying ;  for  example,  a  study  of  the  rate  of  cell-division 
in  a  segmenting  egg  may  teach  us  something  about  the  work  done, 
and  about  the  various  energies  concerned.  But  in  such  cases  the 
action  is,  as  a  rule,  so  homogeneous,  and  the  form  finally  attained 
is  so  definite  and  so  httle  dependent  on  the  time  taken  to  effect 
it,  that  the  specific  rate  of  change,  or  rate  of  growth,  does  not 
enter  into  the  moriihological  problem. 

To  sum  up,  we  may  lay  down  the  following  general  statements. 
The  form  of  organisms  is  a  phenomenon  to  be  referred  in  part 
to  the  direct  action  of  molecular  forces,  in  part  to  a  more  complex 
and  slower  process,  indirectly  resulting  from  chemical,  osmotic 
and  other  forces,  by  which  material  is  introduced  into  the  organism 
and  transferred  from  one  part  of  it  to  another.  It  is  this  latter 
complex  phenomenon  which  we  usually  speak  of  as  "growth." 

*  Cf.  Aristotle,  Phys.  vi,  5,  235  a  11.  eVet  yap  airaaa  Kivrjcm  iv  xP^''Vy  '''''^• 
Bacon  emphasised,  in  like  manner,  the  fact  that  "all  motion  or  natural  action 
is  performed  in  time :  some  more  quickly,  some  more  slowly,  but  all  in  periods 
determined  and  fixed  in  the  nature  of  things.  Even  those  actions  which  seem 
to  be  performed  suddenly,  and  (as  we  say)  in  the  twinkling  of  an  eye,  are  found 
to  admit  of  degree  in  respect  of  duration."     Nov.  Org.  XLVI. 


54    ■  THE   RATE   OF   GROWTH  [ch. 

Every  growing  organism,  and  every  part  of  such  a  growing 
organism,  has  its  own  specific  rate  of  growth,  referred  to  a  particular 
direction.  It  is  the  ratio  between  the  rates  of  growth  in  various 
directions  by  which  we  must  account  for  the  external  forms  of 
all,  save  certain  very  minute,  organisms.  This  ratio  between 
rates  of  growth  in  various  directions  may  sometimes  be  of  a 
simple  kind,  as  when  it  results  in  the  mathematically  definable 
outline  of  a  shell,  or  in  the  smooth  curve  of  the  margin  of  a  leaf. 
It  may  sometimes  be  a  very  constant  one,  in  which  case  the 
organism,  w^hile  growing  in  bulk,  suffers  little  or  no  perceptible 
change  in  form;  but  such  equilibrium  seldom  endures  for  more 
than  a  season,  and  when  the  ratio  tends  to  alter,  then  we  have 
the  phenomenon  of  morphological  "development,"  or  steady  and 
persistent  change  of  form. 

This  elementary  concept  of  Form,  as  determined  by  varying 
rates  of  Growth,  w^as  clearly  apprehended  by  the  mathematical 
mind  of  Haller, — who  had  learned  his  mathematics  of  the  great 
John  BernoulH,  as  the  latter  in  turn  had  learned  his  physiology 
from  the  writings  of  Borelli.  Indeed  it  was  this  very  point,  the 
apparently  unlimited  extent  to  which,  in  the  development  of  the 
chick,  inequalities  of  growth  could  and  did  produce  changes  of 
form  and  changes  of  anatomical  "structure,''  that  led  Haller  to 
surmise  that  the  process  was  actually  without  limits,  and  that  all 
development  was  but  an  unfolding,  or  "evolutio,''  in  which  no 
part  came  into  being  which  had  not  essentially  existed  before  *. 
In  short  the  celebrated  doctrine  of  "preformation"  implied  on  the 
one  hand  a  clear  recognition  of  what,  throughout  the  later  stages 
of  development,  growth  can  do,  by  hastening  the  increase  in  size 
of  one  part,  hindering  that  of  another,  changing  their  relative 
magnitudes  and  positions,  and  altering  their  forms ;  while  on  the 
other  hand  it  betrayed  a  failure  (inevitable  in  those  days)  to 
recognise  the  essential  difference  between  these  movements  of 
masses  and  the  molecular  processes  which  precede  and  accompany 

*  Cf.  (e.g.)  Elem.  Physiol,  ed.  1766,  viii,  p.  114,  '•Ducimur  autem  ad  evolu- 
tionem  potissimum,  quando  a  perfecto  animale  retrorsum  progredimur,  et  incre- 
mentorum  atque  nautationum  seriem  relegimus.  Ita  inveniemus  perfectum  illud 
animal  fuisse  imperfectius,  alterius  figurae  et  fabricae,  et  denique  rude  et  informe : 
et  tamen  idem  semper  animal  sub  iis  diversis  phasibus  fuisse,  quae  absque  uUo 
saltu  perpetuos  parvosque  per  gradus  cohaereant." 


Ill]  THE  DOCTRINE   OF   PREFORMATION  55 

them,  and  which  are  characteristic  of  another  order  of  magni- 
tude. 

By  other  writers  besides  Haller  the  very  general,  though  not 
strictly  universal  connection  between  form  and  rate  of  growth 
has  been  clearly  recognised.  Such  a  connection  is  implicit  in 
those  "proportional  diagrams"  by  which  Diirer  and  some  of  his 
brother  artists  were  wont  to  illustrate  the  successive  changes  of 
form,  or  of  relative  dimensions,  which  attend  the  growth  of  the 
child,  to  boyhood  and  to  manhood.  The  same  connection  was 
recognised,  more  explicitly,  by  some  of  the  older  embryologists, 
for  instance  by  Pander*,  and  appears,  as  a  survival  of  the 
doctrine  of  preformation,  in  his  study  of  the  development  of 
the  chick.  And  long  afterwards,  the  embryological  aspect  of 
the  case  was  emphasised  by  His,  who  pointed  out,  for  instance, 
that  the  various  foldings  of  the  blastoderm,  by  which  the  neural 
and  amniotic  folds  were  brought  into  being,  were  essentially 
and  obviously  the  resultant  of  unequal  rates  of  growth, — of 
local  accelerations  or  retardations  of  growth,^ — in  what  to  begin 
with  was  an  even  and  uniform  layer  of  embryonic  tissue.  If 
we  imagine  a  flat  sheet  of  paper,  parts  of  which  are  caused 
(as  by  moisture  or  evaporation)  to  expand  or  to  contract,  the 
plane  surface  is  at  once  dimpled,  or  "buckled,"  or  folded,  by 
the  resultant  forces  of  expansion  or  contraction :  and  the  various 
distortions  to  which  the  plane  surface  of  the  "germinal  disc"  is 
subject,  as  His  shewed  once  and  for  all,  are  precisely  analogous. 
An  experimental  demonstration  still  more  closely  comparable  to 
the  actual  case  of  the  blastoderm,  is  obtained  by  making  an 
"artificial  blastoderm,"  of  httle  pills  or  pellets  of  dough,  which 
are  caused  to  grow,  with  varying  velocities,  by  the  addition 
of  varying  quantities  of  yeast.  Here,  as  Roux  is  careful  to 
point  outt,  we  observe  that  it  is  not  only  the  growth  of  the 
individual  cells,  but  the  traction  exercised  through  their  mutual 
interconnections,  which  brings  about  the  foldings  and  other  dis- 
tortions of  the  entire  structure. 

*  Beitrdge  zur  Entwickelungsgeschichte  des  Hiihnchens  im  Ei,  p.  40,  1817.  Roux 
ascribes  the  same  views  also  to  Von  Baer  and  to  R.  H.  Lotze  (Allg.  Physiologie, 
p.  353,  1851). 

t  Roux,  Die  Entwickdungsmeclianik,  p.  99,  1905. 


56  THE   RATE   OF   GROWTH  [ch. 

But  this  again  was  clearly  present  to  Haller's  mind,  and  formed 
an  essential  part  of  his  embryological  doctrine.  For  he  has  no 
sooner  treated  of  incrementum,  or  celeritas  incrementi,  than  he 
proceeds  to  deal  with  the  contributory  and  complementary  pheno- 
mena of  expansion,  traction  {adfractio)*,  and  pressure,  and  the 
more  subtle  influences  which  he  denominates  vis  derivationis  et 
revulsio7iis'\ :  these  latter  being  the  secondary  and  correlated 
effects  on  growth  in  one  part,  brought  about,  through  such 
changes  as  are  produced  (for  instance)  in  the  circulation,  by  the 
growth  of  another. 

Let  us  admit  that,  on  the  physiological  side,  Haller's  or  His's 
methods  of  explanation  carry  us  back  but  a  little  way ;  yet  even 
this  little  way  is  something  gained.  Nevertheless,  I  can  well 
remember  the  harsh  criticism,  and  even  contempt,  which  His's 
doctrine  met  with,  not  merely  on  the  ground  that  it  was  inadequate, 
but  because  such  an  explanation  was  deemed  wholly  inappropriate, 
and  was  utterly  disavowed  j.  Hertwig,  for  instance,  asserted  that, 
in  embryology,  when  we  found  one  embryonic  stage  preceding 
another,  the  existence  of  the  former  was,  for  the  embryologist, 
an  all-sufficient  "causal  explanation"  of  the  latter.  "We  consider 
.  (he  says),  that  we  are  studying  and  explaining  a  causal  relation 
when  we  have  demonstrated  that  the  gastrula  arises  by  invagina- 
tion of  a  blastosphere,  or  the  neural  canal  by  the  infolding  of  a 
cell  plate  so  as  to  constitute  a  tube  §."    For  Hertwig,  therefore,  as 

*  Op.  cif.  p.  302,  "  Magnum  hoc  naturae  instrumentum,  etiam  in  corpore 
animato  evolvendo  potenter  operatur;   etc." 

t  Ibid.  p.  306.  "Subtiliora  ista,  et  aliquantum  hypotliesi  mista,  tamen  magnum 
mihi  videntur  speciem  veri  habere." 

{  Cf.  His,  On  the  Principles  of  Animal  Morphology,  Proc.  JR.  S.  E.  xv, 
1888,  p.  294:  "My  own  attempts  to  introduce  some  elementary  mechanical  or 
physiological  conceptions  into  embryology  have  not  generally  been  agreed  to  by 
morphologists.  To  one  it  seemed  ridiculous  to  speak  of  the  elasticity  of  the  germinal 
layers;  another  thought  that,  by  such  considerations,  we  'put  the  cart  before 
the  horse ' :  and  one  more  recent  author  states,  that  we  have  better  things  to  do 
in  embryology  than  to  discuss  tensions  of  germinal  layers  and  similar  questions, 
since  all  explanations  must  of  necessity  be  of  a  phylogenetic  nature.  This  opposition 
to  the  application  of  the  fundamental  principles  of  science  to  embryological  questions 
would  scarcely  be  intelligible  had  it  not  a  dogmatic  background.  No  other  explana- 
tion of  living  forms  is  allowed  than  heredity,  and  any  which  is  founded  on  another 

basis    must    be    rejected To  think  that    hei'edity  will    build  organic  beings 

without  mechanical  means  is  a  piece  of  unscientific  mysticism." 

§  Hertwig,  0.,  Zeit  und  Streitfragen  der  Biologie,  ii,  1897 


Ill]  OF   PHYSICS   AND   EMBRYOLOGY  57 

Roux  remarks,  the  task  of  investigating  a  physical  mechanism  in 
embryology, — "der  Ziel  das  Wirken  zu  erforschen," — has  no 
existence  at  all.  For  Balfour  also,  as  for  Hertwig,  the  mechanical 
or  physical  aspect  of  organic  development  had  httle  or  no  attraction. 
In  one  notable  instance,  Balfour  himself  adduced  a  physical,  or 
quasi-physical,  explanation  of  an  organic  process,  when  he  referred 
the  various  modes  of  segmentation  of  an  ovum,  complete  or  partial, 
equal  or  unequal  and  so  forth,  to  the  varying  amount  or  the 
varying  distribution  of  food  yolk  in  association  with  the  germinal 
protoplasm  of  the  egg*.  But  in  the  main,  Balfour,  hke  all  the 
other  embryologists  of  his  day,  was  engrossed  by  the  problems  of 
phylogeny,  and  he  expressly  defined  the  aims  of  comparative 
embryology  (as  exemphfied  in  his  own  textbook)  as  being  "two- 
fold: (1)  to  form  a  basis  for  Phylogeny.  and  (2)  to  form  a  basis 
for  Organogeny  or  the  origin  and  evolution  of  organsf." 

It  has  been  the  great  service  of  Roux  and  his  fellow-workers 
of  the  school  of  "Entwickelungsmechanik,"  and  of  many  other 
students  to  whose  work  we  shall  refer,  to  try,  as  His  tried  J,  to 
import  into  embryology,  wherever  possible,  the  simpler  concepts 
of  physics,  to  introduce  along  with  them  the  method  of  experiment, 
and  to  refuse  to  be  bound  by  the  narrow  limitations  which  such 
teaching  as  that  of  Hertwig  would  of  necessity  impose  on  the 
work  and  the  thought  and  on  the  whole  philosophy  of  the  biologist. 

Before  we  pass  from  this  general  discussion  to  study  some  of 
the  particular  phenomena  of  growth,  let  me  give  a  single  illustration, 
from  Darwin,  of  a  point  of  view  which  is  in  marked  contrast  to 
Haller's  simple  but  essentially  mathematical  conception  of  Form. 

There  is  a  curious  passage  in  the  Origin  of  Species^,  where 
Darwin  is  discussing  the  leading  facts  of  embryology,  and  in 
particular  Von  Baer's  "law  of  embryonic  resemblance."  Here 
Darwin  says  "We  are  so  much  accustomed  to  see  a  difference  in 

*  Cf.  Roux,  Gesammelte  Ahhandlungen,  ii,  p.  31,  1895. 

t  Treatise  on  Comparative  Embryology,  i,  p.  4,  1881. 

%  Cf.  Fick,  Anal.  Anzeiger,  xxv,  p.  190,  1904. 

§  1st  ed.  p.  444;  6th  ed.  p.  390.  The  student  should  not  fail  to  consult  the 
passage  in  question;  for  there  is  always  a  risk  of  misunderstanding  or  mis- 
interpretation when  one  attempts  to  epitomise  Darwin's  carefully  condensed 
arguments. 


58  THE   RATE   OF   GROWTH  [ch. 

structure  between  the  embryo  and  the  adult,  that  we  are  tempted 
to  look  at  this  dift'erence  as  in  some  necessary  manner  contingent 
on  growth.  But  there  is  no  reason  why,  for  instance,  the  wing  of 
a  hat,  or  the  Jin  of  a  porfoise,  should  not  have  been  sketched  out  with 
all  their  parts  in  proper  proportion,  as  soon  as  any  part  became 
visible.''  After  pointing  out  with  his  habitual  care  various 
exceptions,  Darwin  proceeds  to  lay  down  two  general  principles, 
viz.  "that  shght  variations  generally  appear  at  a  not  very  early 
period  of  hfe,"  and  secondly,  that  "at  whatever  age  a  variation 
first  appears  in  the  parent,  it  tends  to  reappear  at  a  corresponding 
age  in  the  offspring."  He  then  argues  that  it  is  with  nature  as 
with  the  fancier,  who  does  not  care  what  his  pigeons  look  hke 
in  the  embryo,  so  long  as  the  full-grown  bird  possesses  the  desired 
qualities ;  and  that  the  process  of  selection  takes  place  when 
the  birds  or  other  animals  are  nearly  grown  up, — at  least  on  the 
part  of  the  breeder,  and  presumably  in  nature  as  a  general  rule. 
The  illustration  of  these  principles  is  set  forth  as  follows:  "Let 
us  take  a  group  of  birds,  descended  from  some  ancient  form  and 
modified  through  natural  selection  for  different  habits.  Then, 
from  the  many  successive  variations  having  supervened  in  the 
several  species  at  a  not  very  early  age,  and  having  been  inherited 
at  a  corresponding  age,  the  young  will  still  resemble  each  other 
much  more  closely  than  do  the  adults, — just  as  we  have  seen 
with  the  breeds  of  the  pigeon.... Whatever  influence  long-continued 
use  or  disuse  may  have  had  in  modifying  the  hmbs  or  other  parts 
of  any  species,  this  will  chiefly  or  solely  have  affected  it  when 
nearly  mature,  when  it  was  compelled  to  use  its  full  powers  to 
gain  its  own  living;  and  the  effects  thus  produced  will  have  been 
transmitted  to  the  offspring  at  a  corresponding  nearly  mature 
age.  Thus  the  young  will  not  be  modified,  or  will  be  modified 
only  in  a  shght  degree,  through  the  effects  of  the  increased  use  or 
disuse  of  parts."  This  whole  argument  is  remarkable,  in  more 
ways  than  we  need  try  to  deal  with  here ;  but  it  is  especially 
remarkable  that  Darwin  should  -begin  by  casting  doubt  upon  the 
broad  fact  that  a  "difference  in  structure  between  the  embryo 
and  the  adult"  is  "in  some  necessary  manner  contingent  on 
growth";  and  that  he  should  see  no  reason  why  complicated 
structures   of   the   adult   "  should  not   have   been  sketched   out 


Ill]  A  PASSAGE   IN  DARWIN  59 

with  all  their  parts  in  proper  proportion,  as  soon  as  any  part 
became  visible."  It  would  seem  to  me  that  even  the  most 
elementary  attention  to  form  in  its  relation  to  growth  would  have 
removed  most  of  Darwin's  difficulties  in  regard  to  the  particular 
phenomena  which  he  is  here  considering.  For  these  phenomena 
are  phenomena  of  form,  and  therefore  of  relative  magnitude ; 
and  the  magnitudes  in  question  are  attained  by  growth,  proceeding 
with  certain  specific  velocities,  and  lasting  for  certain  long  periods 
of  time.  And  it  is  accordingly  obvious  that  in  any  two  related 
individuals  (whether  specifically  identical  or  not)  the  differences 
between  them  must  manifest  themselves  gradually,  and  be  but 
little  apparent  in  the  young.  It  is  for  the  same  simple  reason 
that  animals  which  are  of  very  dift'erent  sizes  when  adult,  differ 
less  and  less  in  size  (as  well  as  in  form)  as  we  trace  them  back- 
wards through  the  foetal  stages. 

Though  we  study  the  visible  effects  of  varying  rates  of  growth 
throughout  wellnigh  all  the  problems  of  morphology,  it  is  not  very 
often  that  we  can  directly  measure  the  velocities  concerned. 
But  owing  to  the  obvious  underlying  importance  which  the 
phenomenon  has  to  the  morphologist  we  must  make  shift  to  study 
it  where  we  can,  even  though  our  illustrative  cases  may  seem  to 
have  little  immediate  bearing  on  the  morphological  problem*. 

In  a  very  simple  organism,  of  spherical  symmetry,  such  as  the 
single  spherical  cell  of  Protococcus  or  of  Orbulina,  growth  is 
reduced  to  its  simplest  terms,  and  indeed  it  becomes  so  simple 
in  its  outward  manifestations  that  it  is  no  longer  of  special  interest 
to  the  morphologist.  The  rate  of  growth  is  measured  by  the  rate 
of  change  in  length  of  a  radius,  i.e.  V  =  {R'  —  R)/T,  and  from 
this  we  may  calculate,  as  already  indicated,  the  rate  of  growth  in 
terms  of  surface  and  of  volume'.  The  growing  body  remains  of 
constant  form,  owing  to  the  symmetry  of  the  system;  because, 
that  is  to  say,  on  the  one  hand  the  pressure  exerted  by  the  growing 
protoplasm  is  exerted  equally  in  all  directions,  after  the  manner 
of  a  hydrostatic  pressure,  which  indeed  it  actually  is :  while  on 
the  other  hand,  the  "skin"  or  surface  layer  of  the  cell  is  sufficiently 

*  '"In  omni  rerum  naturalium  historia  utile  est  mensuras  definiri  et  numeros," 
Haller,  Elem.  Physiol,  ii,  p.  258,  1760.    Cf.  Hales,  Vegetable  Staficks,  Introduction. 


60  THE   RATE   OF   GROWTH  [ch. 

homogeneous  to  exert  at  every  point  an  approximately  uniform 
resistance.  Under  these  conditions  then,  the  rate  of  growth  is 
uniform  in  all  directions,  and  does  not  affect  the  form  of  the 
organism. 

But  in  a  larger  or  a  more  complex  organism  the  study  of  growth, 
and  of  the  rate  of  growth,  presents  us  with  a  variety  of  problems, 
and  the  whole  phenomenon  becomes  a  factor  of  great  morphological 
importance.  •  We  no  longer  find  that  it  tends  to  be  miiform  in 
all  directions,  nor  have  we  any  right  to  expect  that  it  should. 
The  resistances  which  it  meets  with  will  no  longer  be  uniform. 
In  one  direction  but  not  in  others  it  will  be  opposed  by  the 
important  resistance  of  gravity;  and  within  the  growing  system 
itself  all  manner  of  structural  differences  will  come  into  play, 
setting  up  unequal  resistances  to  growth  by  the  varying  rigidity 
or  viscosity  of  the  material  substance  in  one  direction  or  another. 
At  the  same  time,  the  actual  sources  of  growth,  the  chemical  and 
osmotic  forces  which  lead  to  the  intussusception  of  new  matter, 
are  not  uniformly  distributed ;  one  tissue  or  one  organ  may  well 
manifest  a  tendency  to  increase  while  another  does  not;  a  series 
of  bones,  their  intervening  cartilages,  and  their  surrounding 
muscles,  may  all  be  capable  of  very  different  rates  of  increment. 
The  differences  of  form  which  are  the  resultants  of  these  differences 
in  rate  of  growth  are  especially  manifested  during  that, part  of 
life  when  growth  itself  is  rapid:  when  the  organism,  as  we  say, 
is  undergoing  itfe  develojoment .  When  growth  in  general  has 
become  slow,  the  relative  differences  in  rate  between  different 
parts  of  the  organism  may  still  exist,  and  may  be  made  manifest 
by  careful  observation,  but  in  many,  or  perhaps  in  most  cases,  the 
resultant  change  of  form  does  not  strike  the  eye.  Great  as  are 
the  differences  between  the  rates  of  growth  in  different  parts  of 
an  organism,  the  marvel  is  that  the  ratios  between  them  are  so 
nicely  balanced  as  they  actually  are,  and  so  capable,  accordingly, 
of  keeping  for  long  periods  of  time  the  form  of  the  growing  organism 
all  but  unchanged.  There  is  the  nicest  possible  balance  of  forces 
and  resistances  in  every  part  of  the  complex  body;  and  when 
this  normal  equilibrium  is  disturbed,  then  we  get  abnormal 
growth,  in  the  shape  of  tumours,  exostoses,  and  malformations 
of  every  kind. 


Ill] 


QUETELET'S  ANTHROPOMETRIE 


61 


The  rate  of  growth  in  Man. 

Man  will  serve  us  as  well  as  another  organism  for  our  first 
illustrations  of  rate  of  growth ;  and  we  cannot  do  better  than  go 
for  our  first  data  concerning  him  to  Quetelet's  Anthro'pometrie* ,  an 
epoch-making  book  for  the  biologist.  For  not  only  is  it  packed 
with  information,  some  of  it  still  unsurpassed,  in  regard  to  human 
growth  and  form,  but  it  also  merits  our  highest  admiration  as  the 
first  great  essay  in  scientific  statistics,  and  the  first  work  in  which 
organic  variation  was  discussed  from  the  point  of  view  of  the 
mathematical  theory  of  probabilities. 


Fig.  3.     Curve  of  Growth  in  Man,  from  birth  to  20  yrs  (3) ;  from  Quetelet's  Belgian 
data.     The  upper  curve  of  stature  from  Bowditch's  Boston  data. 

If  the  child  be  some  20  inches,  or  say  50  cm.  tall  at  birth,  and 
the  man  some  six  feet  high,  or  say  180  cm.,  at  twenty,  we  may 
say  that  his  average  rate  of  growth  has  been  (180  —  50)/20  cm.,  or 
6-5  centimetres  per  annum.     But  we  know  very  well  that  this  is 

*  Brussels,  1871.  Cf.  the  same  author's  Physique  sociale,  1835,  and  Lettres 
sur  la  theorie  des  probabilites,  1846.  See  also,  for  the  general  subject,  Boyd,  R., 
Tables  of  weights  of  the  Human  Body,  etc.  Phil.  Trans,  vol.  cli,  1861 ;  Roberts, 
C,  Manual  of  Aiithropometry,  1878;  Daffner,  F.,  Das  Wachsthum  des  Menschen 
(2nd  ed.),  1902,  etc. 


62  THE   RATE   OF   GROWTH  [ch. 

but  a  very  rough  preliminary  statement,  and  that  the  boy  grew 
quickly  during  some,  and  slowly  during  other,  of  his  twenty  years. 
It  becomes  necessary  therefore  to  study  the  phenomenon  of  growth 
in  successive  small  portions ;  to  study,  that  is  to  say,  the  successive 
lengths,  or  the  successive  small  differences,  or  increments,  of 
length  (or  of  weight,  etc.),  attained  in  successive  short  increments 
of  time.  This  we  do  in  the  first  instance  in  the  usual  way,  by 
the  "graphic  method"  of  plotting  length  against  time,  and  so  con- 
structing our  "curve  of  growth."  Our  curve  of  growth,  whether 
of  weight  or  length  (Fig.  3),  has  always  a  certain  characteristic 
form,  or  characteristic  curvature.  This  is  our  immediate  proof  of 
the  fact  that  the  rate  of  growth  changes  as  time  goes  on ;  for  had 
it  not  been  so,  had  an  equal  increment  of  length  been  added  in 
each  equal  interval  of  time,  our  "curve"  would  have  appeared 
as  a  straight  line.  Such  as  it  is,  it  tells  us  not  only  that  the  rate 
of  growth  tends  to  alter,  but  that  it  alters  in  a  definite  and  orderly 
way ;  for,  subject  to  various  minor  interruptions,  due  to  secondary 
causes,  our  curves  of  growth  are,  on  the  whole,  "smooth"  curves. 

The  curve  of  growth  for  length  or  stature  in  man  indicates 
a  rapid  increase  at  the  outset,  that  is  to  say  during  the  quick 
growth  of  babyhood ;  a  long  period  of  slower,  but  still  rapid  and 
almost  steady  growth  in  early  boyhood;  as  a  rule  a  marked 
quickening  soon  after  the  boy  is  in  his  teens,  when  he  comes  to 
"the  growing  age"  ;  and  finally  a  gradual  arrest  of  growth  as  the 
boy  "comes  to  his  full  height,"  and  reaches  manhood. 

If  we  carried  the  curve  further,  we  should  see  a  very  curious 
thing.  We  should  see  that  a  man's  full  stature  endures  but  for 
a  spell;  long  before  fifty*  it  has  begun  to  abate,  by  sixty  it  is 
notably  lessened,  in  extreme  old  age  the  old  man's  frame  is 
shrunken  and  it  is  but  a  memory  that  "he  once  was  tall."  We 
have  already  seen,  and  here  we  see  again,  that  growth  may  have 
a  "negative  value."  The  phenomenon  of  negative  growth  in  old 
age  extends  to  weight  also,  and  is  evidently  largely  chemical  in 
origin :  the  organism  can  no  longer  add  new  material  to  its  fabric 
fast  enough  to  keep  pace  with  the  wastage  of  time.     Our  curve 

*  Dr  Johnson  was  not  far  wrong  in  saying  that  "life  decHnes  from  thirty-five" ; 
though  the  Autocrat  of  the  Breakfast-table,  like  Cicero,  declares  that  "the  furnace 
is  in  full  blast  for  ten  years  longer." 


Ill]  OF  MAN'S   STATURE  63 

of  growth  is  in  fact  a  diagram  of  activity,  or  "time-energy" 
diagram*.  As  the  organism  grows  it  is  absorbing  energy  beyond 
its  daily  needs,  and  accumulating  it  at  a  rate  depicted  in  our 

Stature,  weight,  and  span  of  outstretched  arms. 
{After  Qnetelet,  pp-  193,  346.) 

Stature  in  metres  Weight  in  kgm.  Span  of      %  ratio 

f ^ ^  , '^ -,  arms,      of  stature 

Age        Male        Female      %  F/M       Male       Female    %  F/M       male         to  span 

0  0-500  0-494  98-8  3-2  2-9  90-7  0-496  100-8 

1  0-698  0-690  98-8  9-4  8-8  93-6  0-695  100-4 

2  0-791  0-781  98-7  11-3  10-7  94-7  0-789  100-3 

3  0-864  0-854  98-8  12-4  11-8  95-2  0-863  100-1 

4  0-927  0-915  98-7  14-2  13-0  91-5  0-927  100-0 

5  0-987  0-974  98-7  15-8  14-4  91-1  0-988  99-9 

6  1-046  1-031  98-5  17-2  16-0  93-0  1-048  99-8 

7  1-104  1-087  98-4  19-1  •17-5  91-6  1-107  99-7 

8  1-162  1-142  98-2  20-8  19-1  91-8  1-166  99-6 

9  1-218  1-196  98-2  22-6  21-4  94-7  1-224  99-5 

10  1-273  1-249  98-1  24-5  23-5  95-9  1-281  99-4 

11  1-325  1-301  98-2  27-1  25-6  94-5  1-335  99-2 

12  1-375  1-352  98-3  29-8  29-8  100-0  1-388  99-1 

13  1-423  1-400  98-4  34-4  32-9  95-6  1-438  98-9 

14  1-469  1-446  98-4  38-8  36-7  94-6  1-489  98-7 

15  1-513  1-488  98-3  43-6  40-4  92-7  1-538  99-4 

16  1-554  1-521  97-8  49-7  43-6  87-7  1-584  98-1 

17  1-594  1-546  97-0  52-8  47-3  89-6  1-630  97-9 

18  1-630  1-563  95-9  57-8  49-0  84-8  1-670  97-6 

19  1-655  1-570  94-9  58-0  51-6  89-0  1-705  97-1 

20  1-669  1-574  94-3  60-1  52-3  87-0  1-728  96-6 
25  1-682  1-578  93-8  62-9  53-3  84-7  1-731  97-2 
30  1-686  1-580  93-7  63-7  54-3  85-3  1-766  95-5 
40  1-686  1-580  93-7  63-7  55-2  86-7  1-766  95-5 
50  1-686  1-580  93-7  63-5  56-2  88-4           —  — 
60  1-676  1-571  93-7  61-9  54-3  87-7           —  — 
70  1-660  1-556  93-7  59-5  51-5  86-5           -^  — 
80  1-636  1-534  93-8  57-8  49-4  85-5           —  — 
90  1-610  1-510  93-8  57-8  49-3  85-3           —  — 

curve ;  but  the  time  comes  when  it  accumulates  no  longer,  and  at 
last  it  is  constrained  to  draw  upon  its  dwindling  store.  But  in  part, 
the  slow  decline  in  stature  is  an  expression  of  an  unequal  contest 
between  our  bodily  powers  and  the  unchanging  force  of  gravity, 

*  Joly,  The  Abundance  of  Life,  1915  (1890),  p.  86. 


64  THE   RATE   OF   GROWTH  '         [ch. 

which  draws  us  down  when  we  would  fain  rise  up*.  For  against 
gravity  we  fight  all  our  days,  in  every  movement  of  our  limbs,  in 
every  beat  of  our  hearts ;  it  is  the  indomitable  force  that  defeats 
us  in  the  end,  that  lays  us  on  our  deathbed,  that  lowers  us  to  the 
grave  t- 

Side  by  side  with  the  curve  which  repiesents  growth  in  length, 
or  stature,  our  diagram  shows  the  curve  of  weight  J.  That  this 
curve  is  of  a  very  different  shape  from  the  former  one,  is  accounted 
for  in  the  main  (though  not  wholly)  by  the  fact  which  we  have 
already  dealt  with,  that,  whatever  be  the  law  of  increment  in  a 
Unear  dimension,  the  law  of  increase  in  volume,  and  therefore  in 
weight,  will  be  that  these  latter  magnitudes  tend  to  vary  as 
the  cubes  of  the  linear  dimensions.  This  however  does  not 
account  for  the  change  of  direction,  or  "point  of  inflection" 
which  we  observe  in  the  curve  of  weight  at  about  one  or  two 
years  old,  nor  for  certain  other  differences  between  our  two  curves 
which  the  scale  of  our  diagram  does  not  yet  make  clear.  These 
differences  are  due  to  the  fact  that  the  form  of  the  child  is  altering 
with  growth,  that  other  linear  dimensions  are  varying  somewhat 
differently  from  length  or  stature,  and  that  consequently  the 
growth  in  bulk  or  weight  is  following  a  more  comphcated  law. 

Our  curve  of  growth,  whether  for  weight  or  length,  is  a  direct 
picturs  of  velocity,  for  it  represents,  as  a  connected  series,  the 
successive  epochs  of  time  at  which  successive  weights  or  lengths 
are  attained.  But,  as  we  have  already  in  part  seen,  a  great  part 
of  the  interest  of  our  curve  lies  in  the  fact  that  we  can  see  from 
it,  not  only  that  length  (or  some  other  magnitude)  is  changing, 
but  that  the  rate  of  change  of  magnitude,  or  rate  of  growth,  is 
itseK  changing.  We  have,  in  short,  to  study  the  phenomenon  of 
acceleration:    we  have  begun  by  studying  a  velocity,  or  rate  of 

*  "  Lou  pes,  mestre  de  tout  [Le  poids,  maitre  de  tout],  mestre  senso  vergougnOi 
Que  te  tirasso  en  bus  de  sa  brutalo  pougno,"  J.  H.  Fabre,  Oubreto  prouvenQcdo,  p.  61. 

f  The  continuity  of  the  phenomenon  of  growth,  and  the  natural  passage  from 
the  phase  of  increase  to  that  of  decrease  or  decay,  are  admirably  discussed  by 
Enriques,  in  "  La  morte,"  Biv.  di  Scienza,  1907,  and  in  "  Wachsthum  und  seine 
analytische  Darstellung,"  Biol.  Ceniralbl.  June,  1909.  Haller  (Elem.  vn,  p.  68) 
recognised  decrementum  as  a  phase  of  growth,  not  less  important  (theoretically) 
than  incrementum:    "tristis,  sed  copiosa,  haec  est  materies." 

J  Cf.  (int.  al.),  Friedenthal,  H.,  Das  Wachstum  des  K6rpergewichtes...in 
verschiedenen  Lebensaltem,  Zeit.  f.  allg.  Physiol,  ix,  pp.  487-514,  1909. 


HI]  VELOCITY   AND   ACCELERATION  65 

change  of  magnitude ;  we  must  now  study  an  acceleration,  or 
rate  of  change  of  velocity.  The  rate,  or  velocity,  of  growth  is 
measured  by  th?  slope  of  the  curve ;  where  the  curve  is  steep,  it 
means  that  growth  is  rapid,  and  when  growth  ceases  the  curve 
appears  as  a  horizontal  line.  If  we  can  find  a  means,  then,  of 
representing  at  successive  epochs  the  corresponding  slope,  or 
steepness,  of  the  curve,  we  shall  have  obtained  a  picture  of  the 
rate  of  change  of  velocity,  or  the  acceleration  of  growth.  The 
measure  of  the  steepness  of  a  curve  is  given  by  the  tangent  to 
the  curve,  or  we  may  estimate  it  by  taking  for  equal  intervals 
of  time  (strictly  speaking,  for  each  infinitesimal  interval  of  time) 
the  actual  increment  added  during  that  interval  of  time :  and  in 
practice  this  simply  amounts  to  taking  the  successive  differences 
between  the  values  of  length  (or  of  weight)  for  the  successive 
ages  which  we  have  begun  by  studying.  If  we  then  plot  these 
successive  differences  against  time,  we  obtain  a  curve  each  point 
upon  which  represents  a  velocity,  and  the  whole  curve  indicates 
the  rate  of  change  of  velocity,  and  we  call  it  an  acceleration-curve. 
It  contains,  in  truth,  nothing  whatsoever  that  was  not  implicit 
in  our  former  curve ;  but  it  makes  clear  to  our  eye,  and  brings 
within  the  reach  of  further  investigation,  phenomena  that  were 
hard  to  see  in  the  other  mode  of  representation. 

The  acceleration-curve  of  height,  which  we  here  illustrate,  in 
Fig.  4,  is  very  different  in  form  from  the  curve  of  growth  which 
we  have  just  been  looking  at;  and  it  happens  that,  in  this  case, 
there  is  a  very  marked  difference  between  the  curve  which  we 
obtain  from  Quetelet's  data  of  growth  in  height  and  that  which 
we  may  draw  from  any  other  series  of  observations  known  to  me 
from  British,  French,  American  or  German  writers.  It  begins  (as 
will  be  seen  from  our  next  table)  at  a  very  high  level,  such 
as  it  never  afterwards  attains ;  and  still  stands  too  high,  during 
the  first  three  or  four  years  of  life,  to  be  represented  on  the  scale 
of  the  accompanying  diagram.  From  these  high  velocities  it  falls 
away,  on  the  whole,  until  the  age  when  growth  itself  ceases,  and 
when  the  rate  of  growth,  accordingly,  has,  for  some  years  together, 
the  constant  value  of  nil ;  but  the  rate  of  fall,  or  rate  of  change  of 
velocity,  is  subject  to  several  changes  or  interruptions.  During 
the  first  three  or  four  years  of  life  the  fall  is  continuous  and  rapid, 

T.  G.  5 


66 


THE   RATE   OF   GROWTH 


[CH. 


but  it  is  somewhat  arrested  for  a  wkile  in  childhood,  from  about 
five  years  old  to  eight.  According  to  Quetelet's  data,  there  is 
another  shght  interruption  in  the  falhng  rate  between  the  ages  of 
about  fourteen  and  sixteen ;  but  in  place  of  this  almost  insignificant 
interruption,  the  Enghsh  and  other  statistics  indicate  a  sudden 


70 
mm. 
per 


60 


50 


40 


30- 


<      20 


10 


^ 

\ 

5 

annum 

\ 

;  \    ;' 

- 

\     \ 

,i 

1     '" 

1  Bowditch 

- 

v.,  X 

s 

- 

- 

1 

1 

10 


15 


20 


25 
yrs. 


Fig.  4.     Mean  annual  increments  of  stature  {$),  Belgian  and  American. 

and  very  marked  acceleration  of  growth  beginning  at  about 
twelve  years  of  age,  and  lasting  for  three  or  four  years;  when 
this  period  of  acceleration  is  over,  the  rate  begins  to  fall  again, 
and  does  so  with  great  rapidity.  We  do  not  know  how  far  the 
absence  of  this  striking  feature  in  the  Belgian  curve  is  due  to  the 
imperfections  of  Quetelet's  data,  or  whether  it  is  a  real  and 
significant  feature  in  the  small-statured  race  which  he  investigated. 
Even  apart  from  these  data  of  Quetelet's  (which  seem  to 
constitute  an  extreme  case),   it  is  evident  that  there  are  very 


OF  MAN'S  STATURE 


67 


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68  THE   RATE   OF   GROWTH  [ch. 

marked  differences  between  different  races,  as  we  shall  presently 
see  there  are  between  the  two  sexes,  in  regard  to  the  epochs  of 
acceleration  of  growth,  in  other  words,  in  the  "phase"  of  the 
curve. 

It  is  evident  that,  if  we  pleased,  we  might  represent  the  rate 
of  change  of  acceleration  on  yet  another  curve,  by  constructing  a 
table  of  "second  differences";  this  would  bring  out  certain  very 
interesting  phenomena,  which  here  however  we  must  not  stay  to 
discuss. 

Annual  Increment  of  Weight  in  Man  (kgm.). 
(After  Quetelet,  Anihropometrie,  p.  346*.) 

Increment  Increment 


Age 

Male 

Female 

Age 

Male 

Female 

O-I 

5-9 

5-6 

12-13 

41 

3-5 

1-2 

20 

2-4 

13-14 

40 

3-8 

2-3 

1-5 

1-4 

14-15 

4-1 

3-7 

3-4 

1-5 

1-5 

15-16 

4-2 

3-5 

4-5 

1-9 

1-4 

16-17 

4-3 

3-3 

5-6 

1-9 

1-4 

17-18 

4-2 

30 

6-7 

1-9 

M 

18-19 

3-7 

2-3 

7-8 

1-9 

1-2 

19-20 

1-9 

11 

8-9 

1-9 

20 

20-21 

1-7 

11 

9-10 

1-7 

21 

21-22 

1-7 

0-5 

10-11 

1-8 

2-4 

22-23 

1-6 

0-4 

11-12 

2-0 

3-5 

23-24 

0-9 

-  0-2 

12-13        4-1  3-5  24-25         0-8         -  02 

The  acceleration-curve  for  man's  weight  (Fig.  5),  whether  we 
draw  it  from  Quetelet's  data,  or  from  the  British,  American  and 
other  statistics  of  later  writers,  is  on  the  whole  similar  to  that 
which  we  deduce  from  the  statistics  of  these  latter  writers  in 
regard  to  height  or  stature ;  that  is  to  say,  it  is  not  a  curve  which 
continually  descends,  but  it  indicates  a  rate  of  growth  which  is 
subject  to  important  fluctuations  at  certain  epochs  of  life.  We  see 
that  it  begins  at  a  high  level,  and  falls  continuously  and  rapidly  f 

*  The  values  given  in  this  table  are  not  in  precise  accord  with  those  of  the 
Table  on  p.  63.  The  latter  represent  Quetelet's  results  arrived  at  iii  1835;  the 
former  are  the  means  of  his  determinations  in  1835-40. 

f  As  Haller  observed  it  to  do  in  the  chick  (Elem.  viii,  p.  294):  "Hoc  iterum 
incrementum  miro  ordine  ita  distribuitur,  ut  in  principio  incubationis  maximum 
est:    inde  perpetuo  minuatur." 


Ill] 


OF   BODILY  WEIGHT 


69 


during  the  first  two  or  three  years  of  hfe.  After  a  shght  recovery, 
it  runs  nearly  level  during  boyhood  from  about  five  to  twelve 
years  old;  it  then  rapidly  rises,  in  the  "growing  period"'  of  the 
early  teens,  and  slowly  and  steadily  falls  from  about  the  age  of 
sixteen  onwards.  It  does  not  reach  the  base-line  till  the  man  is 
about  seven  or  eight  and  twenty,  for  normal  increase  of  weight 
continues  during  the  years  when  the  man  is  "filling  out,"  long 
after  growth  in  height  has  ceased ;  but  at  last,  somewhere  about 
thirty,  the  velocity  reaches  zero,  and  even  falls  below  it,  for  then 


S     3 


i4 


year 


Fig.  5.     Mean  annual  increments  of  weight,  in  man  and  woman; 
from  Quetelet's  data. 

the  man  usually  begins  to  lose  weight  a  little.  The  subsequent 
slow  changes  in  this  acceleration-curve  we  need  not  stop  to  deal 
with. 

In  the  same  diagram  (Fig.  5)  I  have  set  forth  the  acceleration- 
curves  in  respect  of  increment  of  weight  for  both  man  and  woman, 
according  to  Quetelet.  That  growth  in  boyhood  and  growth  in 
girlhood  follow  a  very  different  course  is  a  matter  of  common 
knowledge ;  but  if  we  simply  plot  the  ordinary  curve  of  growth, 
or  velocity-curve,  the  difference,  on  the  small  scale  of  our  diagrams, 


70  THE   RATE   OF   C4R0WTH  [ch. 

is  not  very  apparent.  It  is  admirably  brought  out,  however,  in 
the  acceleration-curves.  Here  we  see  that,  after  infancy,  say 
from  three  years  old  to  eight,  the  velocity  in  the  girl  is  steady, 
just  as  in  the  boy,  but  it  stands  on  a  lower  level  in  her  case  than 
in  his :  the  Uttle  maid  at  this  age  is  growing  slower  than  the  boy. 
But  very  soon,  and  while  his  acceleration-curve  is  still  represented 
by  a  straight  hne,  hers  has  begun  to  ascend,  and  until  the  girl 
is  about  thirteen  or  fourteen  it  continues  to  ascend  rapidly. 
After  that  age,  as  after  sixteen  or  seventeen  in  the  boy's  case,  it 
begins  to  descend.  In  short,  throughout  all  this  period,  it  is  a  very 
similar  curve  in  the  tw^o  sexes ;  but  it  has  its  notable  differences, 
in  ampUtude  and  especially  in  fliase.  Last  of  all,  we  may  notice 
that  while  the  acceleration-curve  falls  to  a  negative  value  in  the 
male  about  or  even  a  httle  before  the  age  of  thirty  years,  this 
does  not  happen  among  women.  They  continue  to  grow  in 
weight,  though  slowly,  till  very  much  later  in  hfe ;  until  there 
comes  a  final  period,  in  both  sexes  alike,  during  which  weight, 
and  height  and  strength  all  ahke  diminish. 

From  certain  corrected,  or  "typical"  values,  given  for  American  children 
by  Boas  and  Wissler  {I.e.  p.  42),  we  obtain  the  following  still  clearer  comparison 
of  the  armual  increments  of  stature  in  boys  and  girls :  the  typical  stature  at 
the  commencement  of  the  period,  i.e.  at  the  age  of  eleven,  being  135-1  cm. 
and  136-9  cm.  for  the  boys  and  gu'ls  respectively,  and  the  annual  incx'ements 
being  as  follows : 

Age  12  13  14  15  16  17  18  19  20 

Boys  (cm.)  4-1  6-3  8-7  7-9  5-2  3-2  1-9  0-9  0-3 

Girls  (cm.)  7-5  7-0  4-6  2-1  0-9  0-4  0-1  0-0  0-0 

Difference  -3-4  -07  4-1  58  43  2-8  18  09  0-3 

The  result  of  these  differences  (which  are  essentially  phase- 
difierences)  between  the  two  sexes  in  regard  to  the  velocity  of 
growth  and  to  the  rate  of  change  of  that  velocity,  is  to  cause  the 
ratio  between  the  weights  of  the  two  sexes  to  fluctuate  in  a  some- 
what complicated  manner.  At  birth  the  baby-girl  weighs  on  the 
average  nearly  10  per  cent,  less  than  the  boy.  Till  about  two 
years  old  she  tends  to  gain  upon  him,  but  she  then  loses  again 
until  the  age  of  about  five ;  from  five  she  gains  for  a  few  years 
somewhat  rapidly,  and  the  girl  of  ten  to  twelve  is  only  some 
3  per  cent,  less  in  weight  than  the  boy.     The  boy  in  his  teens  gains 


Ill]  OF   BODILY   WEIGHT  71 

steadily,  and  the  young  woman  of  twenty  is  nearly  15  per  cent, 
lighter  than  the  man.  This  ratio  of  difference  again  slowly 
diminishes,  and  between  fifty  and  sixty  stands  at  about  12  per 
cent.,  or  not  far  from  the  mean  for  all  ages;  but  once  more  as 
old  age  advances,  the  difference  tends,  though  very  slowly,  to 
increase  (Fig.  6). 

While  careful  observations  on  the  rate  of  growth  in  other 
animals  are  somewhat  scanty,  they  tend  to  show  so  far  as  they 
go  that  the  general  features  of  the  phenomenon  are  always  much 
the  same.  Whether  the  animal  be  long-lived,  as  man  or  the 
elephant,  or  short-lived,  like  horse  or  dog,  it  passes  through  the 


80       90 
years 

Fig.  6.     Percentage  ratio,  throughout  life,  of  female  weight  to  male; 
from  Quetelet's  data. 

same  phases  of  growth*.  In  all  cases  growth  begins  slowly;  it 
attains  a  maximum  velocity  early  in  its  course,  and  afterwards 
slows  down  (subject  to  temporary  accelerations)  tow^ards  a  point 
where  growth  ceases  altogether.  But  especially  in  the  cold- 
blooded animals,  such  as  fishes,  the  slowing-down  period  is  very 
greatly  protracted,  and  the  size  of  the  creature  would  seem  never 
actually  to  reach,  but  only  to  approach  asymptotically,  to  a 
maximal  limit. 

The  size  ultimately  attained  is  a  resultant  of  the  rate,  and  of 

*  There  is  a  famous  passage  in  Lucretius  (v,  883)  where  he  compares  the  course 
of  life,  or  rate  of  growth,  in  the  horse  and  his  boyish  master :  Principio  circum 
trihus  actis  imfiger  annis  Floret  equus,  puer  hautquaquam,  etc. 


72  THE   RATE   OF   GROWTH  [ch. 

the  duration,  of  growth.  It  is  in  the  main  true,  as  Minot  has 
said,  that  the  rabbit  is  bigger  than  the  guinea-pig  because  he 
grows  the  faster ;  but  that  man  is  bigger  than  the  rabbit  because 
he  goes  on  growing  for  a  longer  time. 


In  ordinary  physical  investigations  dealing  with  velocities,  as 
for  instance  with  the  course  of  a  projectile,  we  pass  at  once  from 
the  study  of  acceleration  to  that  of  momentum  and  so  to  that  of 
force;  for  change  of  momentum,  which  is  proportional  to  force, 
is  the  product  of  the  mass  of  a  body  into  its  acceleration  or  change 
of  velocity.  But  we  can  take  no  such  easy  road  of  kinematical 
investigation  in  this  case.  The  "velocity"  of  growth  is  a  very 
different  thing  from  the  "velocity"  of  the  projectile.  The  forces 
at  work  in  our  case  are  not  susceptible  of  direct  and  easy  treatment ; 
they  are  too  varied  in  their  nature  and  too  indirect  in  their  action 
for  us  to  be  justified  in  equating  them  directly  with  the  mass  of 
the  growing  structure. 

It  was  apparently  from  a  feeling  that  the  velocity  of  growth  ought  in  some 
way  to  be  equated  with  the  mass  of  the  growing  structure  that  Minot*  intro- 
duced a  curious,  and  (as  it  seems  to  me)  an  unhappy  method  of  representing 
growth,  in  the  form  of  what  he  called  "  percentage- curves  "  ;  a  method  which  has 
been  followed  by  a  number  of  other  writers  and  experimenters.  Minot's  method 
was  to  deal,  not  with  the  actual  increments  added  in  successive  periods,  such 
as  years  or  days,  but  with  these  increments  represented  as  percentages  of  the 
amount  which  had  been  reached  at  the  end  of  the  former  period.  For  instance, 
taking  Quetelet's  values  for  the  height  in  centimetres  of  a  male  infant  from 
birth  to  four  years  old,  as  follows: 


Years 

0 

1 

9 

3 

4 

cm. 

50-0 

69-8 

791 

86-4 

92-7 

Jlinot  would  state  the  percentage  growth  in  each  of  the  four  annual  periods 
at  39-6,  13-3,  9-6  and  7-3  per  cent,  respectively. 

Now  when  we  plot  actual  length  against  time,  we  have  a  perfectly  definite 
thing.  When  we  differentiate  this  LjT ,  we  have  dL/dT ,  which  is  (of  course) 
velocity;  and  from  this,  by  a  second  differentiation,  we  obtain  d^L/dT-,  that 
is  to  say,  the  acceleration. 

*  Minot,  C.  S.,  Senescence  and  Rejuvenation,  Journ.  of  Physiol,  xn,  pp.  97— 
153,  1891;  The  Problem  of  Age,  Growth  and  Death,  Poj).  Science  Monthly 
(June-Dec),  1907. 


Ill]      OF   PRE-NATAL   AND   POST-NATAL   GROWTH       73 

But  when  you  take  percentages  of  y,  you  are  determining  dyly,  and  when 
you  plot  this  against  dx,  you  have 

dijly  dy  1    dy 

dx  y .  ax  y    ax 

that  is  to  say,  you  are  multiplying  the  thing  you  wish  to  represent  by  another 
quantity  which  is  itself  continually  varying ;  and  the  result  is  that  you  are 
dealing  with  something  very  much  less  easily  grasped  by  the  mind  than  the 
original  factors.  Professor  Minot  is,  of  course,  dealing  with  a  perfectly 
legitimate  function  of  x  and  y ;  and  his  method  is  practically  tantamount  to 
plotting  log  y  against  x,  that  is  to  say,  the  logarithm  of  the  increment  against 
the  time.  This  could  only  be  defended  and  justified  if  it  led  to  some  simple 
result,  for  instance  if  it  gave  us  a  straight  line,  or  some  other  simpler  curve 
than  our  usual  curves  of  growth.  As  a  matter  of  fact,  it  is  manifest  that  it 
does  nothing  of  the  kind. 

Pre-natal  and  fost-natal  groivtli. 

In  the  acceleration-curves  which  we  have  shown  above  (Figs. 
2,  3),  it  will  be  seen  that  the  curve  starts  at  a  considerable  interval 
-  from  the  actual  date  of  birth ;  for  the  first  two  increments  which 
we  can  as  yet  compare  with  one  another  are  those  attained  during 
the  first  and  second  complete  years  of  life.  Now^  we  can  in  many 
cases  "interpolate"  with  safety  between  known  points  upon  a 
curve,  but  it  is  very  much  less  safe,  and  is  not  very  often  justifiable 
(at  least  until  we  understand  the  physical  principle  involved,  and 
its  mathematical  expression),  to  "extrapolate"  beyond  the  limits 
of  our  observations.  In  short,  we  do  not  yet  know  whether  our 
curve  continued  to  ascend  as  we  go  backwards  to  the  date  of 
birth,  or  whether  it  may  not  have  changed  its  direction,  and 
descended,  perhaps,  to  zero-value.  In  regard  to  length,  or 
stature,  however,  we  can  obtain  the  requisite  information  from 
certain  tables  of  Riissow's*,  who  gives  the  stature  of  the  infant 
month  by  month  during  the  first  year  of  its  life,  as  follows : 

Age  in  months  0       12345       67      8       9     10     11      12 

Length  in  cm.  (50)    54    58    60   62    64    65    66    67-5   68    69    70-5    72 

[Differences  (in  cm.)  4      4      2      2      2      1       1    1-5       -5     1     1-5     1-5] 

If  we  multiply  these   )nonthlij  differences,   or  mean  monthly 
velocities,  by  12,  to  bring  them  into  a  form  comparable  with  the 

*  Quoted  in  Vierordt's  Anatomische...Daten  und  Tabellen,  1906.  p.  13. 


74 


THE   RATE   OF   GROWTH 


[CH. 


anmial  velocities  already  represented  on  our  acceleration-curves, 
we  shall  see  that  the  one  series  of  observations  joins  on  very  well 
with  the  other ;  and  in  short  we  see  at  once  that  our  acceleration- 
curve  rises  steadily  and  rapidly  as  we  pass  back  towards  the  date 
of  birth. 

But  birth  itself,  in  the  case  of  a  viviparous  animal,  is  but  an 
unimportant  epoch  in  the  history  of  growth.  It  is  an  epoch  whose 
relative  date  varies  according  to  the  particular  animal:    the  foal 


0 

cms. 


70 
60 
50 
40 
30 
20 
10 


- 

^^/-^ 

/ 

1 

"^ 

2        4 


6         8       10       12       14      16       18       20    22 

months 


Fig.  7.     Curve  of  growth  (in  length  or  stature)  of  child,  before  and  after 
birth.     (From  His  and  Riissow's  data.) 


and  the  lamb  are  born  relatively  later,  that  is  to  say  when  develop- 
ment has  advanced  much  farther,  than  in  the  case  of  man ;  the 
kitten  and  the  puppy  are  born  earher  and  therefore  more  helpless 
than  we  are;  and  the  mouse  comes  into  the  world  still  earher 
and  more  inchoate,  so  much  so  that  even  the  little  marsupial  is 
scarcely  more  unformed  and  embryonic.  In  all  these  cases  ahke, 
we  must,  in  order  to  study  the  curve  of  growth  in  its  entirety, 
take  full  account  of  prenatal  or  intra-uterine  growth. 


Ill]      OF   PRE-NATAL   AND   POST-NATAL   GROWTH       75 

According  to  His*,  the  following  are  the  mean  lengths  of  the 
unborn  human  embryo,  from  month  to  month. 

Months  01         2345         6         7         89        10 

(Birth) 
Length  in  mm.  0  7-5   40   84  162  275  352  402  443  472  490) 

500) 

Increment  per     —   75     325    44       78     113       77       .50       41       29       18 1 

month  in  mm.  28  I 

These  data  link  on  very  well  to  those  of  Riissow,  which  we 
have  just  considered,  and  (though  His's  measurements  for  the 


0  2        4         6 

Fig.  8.     Mean  monthly  increments  of  length  or  stature  of  child  (in  cms.) 


8       10       12      14       16       18      20    22 

months 


pre-natal  months  are  more  detailed  than  are  those  of  Riissow  for 
the  first  year  of  post-natal  life)  we  may  draw  a  continuous  curve  of 
growth  (Fig.  7)  and  curve  of  acceleration  of  growth  (Fig.  8)  for  the 
combined  periods.  It  will  at  once  be  seen  that  there  is  a  "point 
of  inflection"  somewhere  about  the  fifth  month  of  intra-uterine 
life  f  :  up  to  that  date  growth  proceeds  with  a  continually  increasing 

*   Unsere  KorjKrforrn,  Leipzig,  1874. 

t  No  such  pomt  of  inflection  appears  in  the  curve  of  weight  according  to 
C.  M.  Jackson's  data  (On  the  Prenatal  Growth  of  the  Human  Body,  etc.,  Arner. 
Journ.  of  Anat.  ix.  1909,  jip.  126    156),  nor  in  those  quoted  by  him  from  Ahlfeld, 


76 


THE   RATE   OF   GROWTH 


[CH. 


velocity;  but  after  that  date,  though  growth  is  still  rapid,  its 
velocity  tends  to  fall  away.  There  is  a  shght  break  between  our 
two  separate  sets  of  statistics  at  the  date  of  birth,  while  this  is 
the  very  epoch  regarding  which  we  should  particularly  like  to 
have  precise  and  continuous  information.  Undoubtedly  there  is 
a  certain  shght  arrest  of  growth,  or  diminution  of  the  rate  of 
growth,  about  the  epoch  of  birth :    the  sudden  change  in  the 


- 

y    Length 

- 

•  /                                                < 

/                                                  ^j 

/                                                    "^ 

/                                                    "^ 

/                                                      '*- 

/                                                         ° 

-- 

/                                                           ^ 

/                                                             '^ 

/                                                                 o 

,. 

'                                                                  .'^ 

/ 

ki 

_ 

/ 

,'''    ~^"^v 

>" 

^»* 

/      '^ 

■~^^^^  Acceleration 

yfi-^ 

"  t^^ 

^-xC'  ' 

^~~^~^ 

^>- 

1                  1 

1                        1            .         "" 

500 
mm. 


400 


300 


200^ 


100 


0  2  4  6  8  10 

months 

Fig.  9.     Curve  of  pre-natal  growth  (length  or  stature)  of  child;    and 
correspondmg  curve  of  mean  monthly  increments  (mm.). 

Fehling  and  others.  But  it  is  plain  that  the  very  rapid  increase  of  the  monthly 
weights,  approximately  in  the  ratio  of  the  cubes  of  the  corresponding  lengths, 
would  tend  to  conceal  any  such  breach  of  continuity,  unless  it  happened  to  be  very 
marked  indeed.  Moreover  in  the  case  of  Jackson's  data  (and  probably  also  in 
the  others)  the  actual  age  of  the  embryos  was  not  determined,  but  was  estimated 
from  their  lengths.  The  following  is  Jackson's  estimate  of  average  weights  at 
intervals  of  a  limar  month : 

Months      0123456         7  8  9         10 

Wt  in  gms.    -0     -04     3     36     120     330     600     1000     1500     2200     3200 


Ill]      OF   PRE-NATAL   AND   POST-NATAL   GROWTH       77 

method  of  nutrition  has  its  inevitable  effect;  but  this  shght 
temporary  set-back  is  immediately  followed  by  a  secondary,  and 
temporary,  acceleration. 

It  is  worth  our  while  to  draw  a  separate  curve  to  illustrate  on 
a  larger  scale  His's  careful  data  for  the  ten  months  of  pre-natal 
life  (Fig.  9).  We  see  that  this  curve  of  growth  is  a  beautifully 
regular  one,  and  is  nearly  symmetrical  on  either  side  of  that  point 
of  inflection  of  which  we  have  already  spoken;  it  is  a  curve  for 
which  we  might  well  hope  to  find  a  simple  mathematical  expression. 
The  acceleration-curve  shown  in  Fig.  9  together  with  the  pre-natal 


20   22    24    26    28  30 
days 

Fig.   10.     Curve  of  growth  of  bamboo  (from  Ostwald,  after  Kraus). 

I 

curve  of  growth,  is  not  taken  directly  from  His's  recorded  data, 

but  is  derived  from  the  tangents  drawn  to  a  smoothed  curve, 
corresponding  as  nearly  as  possible  to  the  actual  curve  of  growth : 
the  rise  to  a  maximal  velocity  about  the  fifth  month  and  the 
subsequent  gradual  fall  are  now  demonstrated  even  more  clearly 
than  before.  In  Fig.  10,  which  is  a  curve  of  growth  of  the 
bamboo*,  we  see  (so  far  as  it  goes)  the  same  essential  features, 

*  G.  Kraus  (after  Wallich-Martius),  Ann.  du  Jardin  hot.  dc  Buitenzonj,  xii,  1, 
1894,  p.  210.     Cf.  W.  Ostwald,  Zeitliche  Eiyenschaften,  etc.  p.  56. 


78  THE   RATE   OF   GROWTH  [ch. 

the  slow  beginning,  the  rapid  increase  of  velocity,  the  point  of 
inflection,  and  the  subsequent  slow  negative  acceleration  *. 

Variability  and  Correlation  of  Growth. 

The  magnitudes  and  velocities  which  we  are  here  deahng  with 
are,  of  course,  mean  values  derived  from  a  certain  number,  some- 
times a  large  number,  of  individual  cases.  But  no  statistical 
account  of  mean  values  is  complete  unless  we  also  take  account 
of  the  amount  of  variability  among  the  individual  cases  from  which 
the  mean  value  is  drawn.  To  do  this  throughout  would  lead  us 
into  detailed  investigations  which  he  far  beyond  the  scope  of  this 
elementary  book ;  but  we  ^  may  very  briefly  illustrate  the  nature 
of  the  process,  in  connection  with  the  phenomena  of  growth 
which  we  have  just  been  studying. 

It  was  in  connection  with  these  phenomena,  in  the  case  of 
man,  that  Quetelet  first  conceived  the  statistical  study  of  variation, 
on  hnes  which  were  afterwards  expounded  and  developed  by 
Galton,  and  which  have  grown,  in  the  hands  of  Karl  Pearson  and 
others,  into  the  modern  science  of  Biometrics. 

When  Quetelet  tells  us,  for  instance,  that  the  mean  stature 
of  the  ten-year  old  boy  is  1-273  metres,  this  implies,  according  to 
the  law  of  error,  or  law  of  probabihties,  that  all  the  individual 
measurements  of  ten-year-old  boys  group  themselves  iti  an  orderly 
ivay,  that  is  to  say  according  to  a  certain  definite  law,  about  this 
mean  value  of  1-273.  When  these  individual  measurements  are 
grouped  and  plotted  as  a  curve,  so  as  to  show  the  number  of 
individual  cases  at  each  individual  length,  we  obtain  a  characteristic 
curve  of  error  or  curve  of  frequency;  and  the  "spread"  of  this 
curve  is  a  measure  of  the  amount  of  variabihty  in  this  particular 
case.  A  certain  mathematical  measure  of  this  "spread,"  as 
described  in  works  upon  statistics,  is  called  the  Index  of  Variabihty, 
or  Standard  Deviation,  and  is  usually  denominated  by  the  letter  cr. 
It  is  practically  equivalent  to  a  determination  of  the  point  upon 
the  frequency  curve  where  it  changes  its  curvature  on  either  side 
of  the  mean,  and  where,  from  being  concave  towards  the  middle 
line,  it  spreads  out  to  be  convex  thereto.     When  we  divide  this 

*  Cf.  Chodat,  R.,  et  Monnier.  A.,  Sur  la  courbe  de  croissance  des  vegetaux. 
Bull.  Herb.  Boissier  (2),  v,  pp.  615,  616,  1905. 


Ill]  ITS   VARIABILITY  79 

value  by  the  mean,  we  get  a  figure  which  is  independent  of 
any  particular  units,  and  which  is  called  the  Coefl&cient  of  Varia- 
bility. (It  is  usually  multiphed  by  100,  to  make  it  of  a  more 
convenient  amount ;  and  we  may  then  define  this  coefficient,  C, 
as  =  a/M  X  100.) 

In  regard  to  the  growth  of  man,  Pearson  has  determined  this 
coefficient  of  variabihty  as  follows :  in  male  new-born  infants, 
the  coefficient  in  regard  to  weight  is  15-66,  and  in  regard  to 
stature,  6-50 ;  in  male  adults,  for  weight  10-83,  and  for  stature,  3'66. 
The  amount  of  variabihty  tends,  therefore,  to  decrease  with 
growth  or  age. 

Similar  determinations  have  been  elaborated  by  Bowditqh,  by 
Boas  and  Wissler,  and  by  other  writers  for  intermediate  ages, 
especially  from  about  five  years  old  to  eighteen,  so  covering  a 
great  part  of  the  whole  period  of  growth  in  man*. 

Coefficient  of  Variability  (ujM  x  100)  in  Man,   at  various  ages. 

Age                                                        5  6  ■?                8  9 

Stature  (Bowditch)          ...  4-76  4-60  4-42  4-49  4-40 

„       (Boas  and  Wissler)  4-15  4-14  4-22  4-37  4-33 

Weight  (Bowditch)           ...  11-56  10-28  11-08  9-92  11-04 

Age  10  11  12  13  14 

Stature  (Bowditch)  ...  4-55  4-70  4-90  5-47  5-79 

„       (Boas  and  Wissler)  4-36  4-54  4-73  5-16  5-57 

Weight  (Bowditch)  ...  11-60  11-76  13-72  13-60  16-80 

Age  15  16  17  18 

Stature  (Bowditch)  ....  5-57  4-50  4-55  3-69 

„       (Boas  and  Wissler)  5-50  4-69  4-27  3-94 

Weight  (Bowditch)  ...  15-32  13-28  12-96  10-40 

The  result  is  very  curious  indeed.  We  see,  from  Fig.  11, 
that  the  curve  of  variabihty  is  very  similar  to  what  we  have  called 
the  acceleration-curve  (Fig.  4) :  that  is  to  say,  it  descends  when  the 
rate  of  growth  diminishes,  and  rises  very  markedly  again  when,  in 
late  boyhood,  the  rate  of  growth  is  temporarily  accelerated.     We 

*  Cf.  Fr.  Boas,  Growth  of  Toronto  Children,  Bep.  of  U.S.  Comm.  of  Education, 
1896-7,  pp.  1541-1599,  1898;  Boas  and  Clark  Wissler,  Statistics  of  Growth, 
Education  Re}].  1904,  pp.  25-132,  1906 ;  H.  P.  Bowditch,  Rep.  Mass.  State  Board 
of  Health,  1877 ;  K.  Pearson,  On  the  Magnitude  of  certain  coefficients  of  Correlation 
in  Man,  Pr.  R.  S.  lxvi,  1900. 


80 


THE   RATE   OF   GROWTH 


[CH. 


see,  in  short,  that  the  amount  of  variability  in  stature  or  in  weight 
is  a  function  of  the  rate  of  growth  in  these  magnitudes,  though 
we  are  not  yet  in  a  position  to  equate  the  terms  precisely,  one  witli 
another. 

If  we  take  not  merely  the  variability  of  stature  or  weight  at 
a  given  age,  but  the  variability  of  the  actual  successive  increments 
in  each  yearly  period,  we  see  that  this  latter  coefficient  of  variability 
tends  to  increase  steadily,  and  more  and  more  rapidly,  within 


sfi 


o 


Fig.  11. 


Coefficients  of  variability  of  stature  in  Man  {^).  from  Boas 
and  Wissler's  data. 


the  limits  of  age  for  w^hich  we  have  information ;  and  this  pheno- 
menon is,  in  the  main,  easy  of  explanation.  For  a  great  part  of 
the  difference,  in  regard  to  rate  of  growth,  between  one  individual 
and  another  is  a  difference  of  phase, — a  difference  in  the  epochs 
of  acceleration  and  retardation,  and  finally  in  the  epoch  when 
growth  comes  to  an  end.  And  it  follows  that  the  variability  of 
rate  will  be  more  and  more  marked,  as  we  approach  and  reach 
the  period  when  some  individuals  still  continue,  and  others  have 
already   ceased,    to    grow.     In    the   following   epitomised   table, 


mj  ITS  VARIABILITY  81 

I  have  taken  Boas's  determinations  of  variability  (cr)  {op.  cit. 
p.  1548),  converted  them  into  the  corresponding  coefficients  of 
variability  {a[M  x  100),  and  then  smoothed  the  resulting  numbers. 

Coefficients  of  Variability  in  Annual  Increment  of  Stature. 

Age  7  8  9         10        11        12        13        14        15      § 

Boys       17-3     15-8     18-6     191     21-0    24-7     29-0     36-2     46-1 
Girls       17-1     17-8     19-2     22-7     25-9     29-3     37-0    44-8      — 

The  greater  variability  of  annual  increment  in  the  girls,  as 
compared  with  the  boys,  is  very  marked,  and  is  easily  explained 
by  the  more  rapid  rate  at  which  the  girls  run  through  the  several 
phases  of  the  phenomenon. 

Just  as  there  is  a  marked  difference  in  "phase"  between  the  growth- 
curves  of  the  two'sexes,  that  is  to  say  a  difference  in  the  periods  when  growth 
is  rapid  or  the  reverse,  so  also,  within  each  sex,  will  there  be  room  for  similar, 
but  individual  phase-differences.  Thus  we  may  have  children  of  accelerated 
development,  who  at  a  given  epoch  after  birth  are  both  rapidly  growing  and 
already  "big  for  their  age";  and  others  of  retarded  development  who  are 
comparatively  small  and  have  not  reached  the  period  of  acceleration  which, 
in  greater  or  less  degree,  will  come  to  them  in  turn.  In  other  words,  there 
must  under  such  circumstances  be  a  strong  positive  "coefficient  of  correlation" 
between  stature  and  rate  of  growth,  and  also  between  the  rate  of  growth  in 
one  year  and  the  next.  But  it  does  not  by  any  means  follow  that  a  child  who 
is  precociously  big  will  continue  to  grow  rapidly,  and  become  a  man  or  woman 
of  exceptional  stature.  On  the  contrary,  when  in  the  case  of  the  precocious 
or  "accelerated"  children  growth  has  begun  to  slow  down,  the  backward 
ones  may  still  be  growing  rapidly,  and  so  making  up  (more  or  less  completely) 
to  the  others.  In  other  words,  the  period  of  high  positive  correlation  between 
stature  and  increment  will  tend  to  be  followed  by  one  of  negative  correlation. 
This  interesting  and  important  point,  due  to  Boas  and  Wissler*,  is  confirmed 
by  the  following  table : — 

Correlation  of  Stature  and  Increment  in  Boys  and  Girls. 
{From  Boas  and  Wissler.) 


Age 

6          7           8           9           10         11         12         13         14         15 

Stature         (B) 

112-7    115-5    123-2    127-4    133-2    136-8    142-7    147-3    155-9    162-2 

(G) 

111-4    117-7    121-4    127-9    131-8    136-7    144-6    149-7    153-8    157-2 

Increment    (B) 

5-7        5:3        4-9        5-1        50        4-7        5-9        7-5        6-2        5-2 

(G) 

5-9        5-5        5-5        5-9        6-2        7-2        6-5        5-4        3-3        1-7 

Correlation  (B) 

-25        -11        -08        -25        -18        -18        -48        -29     --42     --44 

(G) 

-44        -14        -24        -47        -18     -18     --42     --39     -63        -11 

I.e.  p.  42,  and  other  papers  there  quoted. 


82  THE  RATE   OF   GROWTH  [ch. 

A  minor,  but  very  curious  point  brought  out  by  the  same  investigators 
is  that,  if  instead  of  stature  we  deal  with  lieight  in  the  sitting  posture  (or, 
practically  speaking,  with  length  of  trunk  or  back),  then  the  correlations 
between  this  height  and  its  annual  increment  are  throughout  negative.  In 
other  words,  there  would  seem  to  be  a  general  tendency  for  the  long  trunks 
to  grow  slowly  throughout  the  whole  period  under  investigation.  It  is  a 
well-known  anatomical  fact  that  tallness  is  in  the  main  due  not  to  length  of 
body  but  to  length  of  limb. 

The  whole  phenomenon  of  variabiUty  in  regard  to  magnitude 
and  to  rate  of  increment  is  in  the  highest  degree  suggesti.ve : 
inasmuch  as  it  helps  further  to  remind  and  to  impress  upon  us 
that  specific  rate  of  growth  is  the  real  physiological  factor  which 
we  want  to  get  at,  of  which  specific  magnitude,  dimensions  and 
form,  and  all  the  variations  of  these,  are  merely  the  concrete  and 
visible  resultant.  But  the  problems  of  variabihty,  though  they 
are  intimately  related  to  the  general  problem  of  growth,  carry  us 
very  soon  beyond  our  present  limitations. 

Rate  of  grow ih  in  other  organisms'^. 

Just  as  the  human  curve  of  growth  has  its  shght  but  well- 
marked  interruptions,  or  variations  in  rate,  coinciding  with  such 
epochs  as  birth  and  puberty,  so  is  it  with  other  animals,  and  this 
phenomenon  is  particularly  striking  in  the  case  of  animals  which 
undergo  a  regular  metamorphosis. 

In  the  accompanying  curve  of  growth  in  weight  of  the  mouse 
(Fig.  12),  based  on  W.  Ostwald's  observations  f,  we  see  a  distinct 
slackening  of  the  rate  when  the  mouse  is  about  a  fortnight  old, 
at  which  period  it  opens  its  eyes  and  very  soon  afterwards  is 
weaned.  At  about  six  weeks  old  there  is  (  nother  well-marked 
retardation  of  growth,  following  on  a  very  rapid  period,  and 
coinciding  with  the  epoch  of  puberty. 

*  See,  for  an  admirable  resume  of  facts,  Wolfgang  Ostwald,  Ueber  die  ZeitUche 
Eigenschaften  der  Entwickelungsvorgdnge  (71  pp.),  Leip/.ig,  1908  (Roux's  Vortrdge, 
Heft  v) :  to  which  work  I  am  much  indebted.  A  long  list  of  observations  on  the 
growth-rate  of  various  animals  is  also  given  by  H.  Przibram,  Exp.  Zoologie,  1913, 
pt  IV  ( Vitalitdt),  pp.  85-87. 

f  Cf .  St  Loup,  Vitesse  de  croissanoe  chez  les  Souris,  Bull.  Soc.  Zool.  Fr.  xvm, 
242,  1893;  Robertson,  Arch.  f.  Entwickelung-smech.  xxv,  p.  587,  1908;  Donaldson. 
Boas  Memorial  Volume,  New  York,  1906. 


Ill 


ITS   PERIODIC   RETARDATION 


83 


Fig.  13  shews  the  curve  of  growth  of  the  silkworm*,  during  its 
whole  larv  1  hfe,  up  to  the  time  of  its  entering  the  chrysalis  stage. 

The  silkworm  moults  four  times,  at  intervals  of  about  a  week, 
the  first  moult  being  on  the  sixth  or  seventh  day  after  hatching. 
A  distinct  retardation  of  growth  is  exhibited  on  our  curve  in  the 
case  of  the  third  and  fourth  moults ;  while  a  similar  retardation 
accompanies  the  first  and  second  moults  also,  but  the  scale  of 
our  diagram  does  not  render  it  visible.  When  the  worm  is  about 
seven  weeks  old,  a  remarkable  process  of  "purgation"  takes  place, 


0         5 
Fig.   12.     Growth  in  weight  of  Mouse.     (After  W.  Ostwald.) 


as  a  preliminary  to  entering  on  the  pupal,  or  chrysaHs,  stage; 
and  the  great  and  sudden  loss  of  weight  which  accompanies  this 
process  is  the  most  marked  feature  of  our  curve. 

The  rate  of  growth  in  the  tadpole  |  (Fig.  14)  is  hkewise  marked 
by  epochs  of  retardation,  and  finally  by  a  sudden  and  drastic 
change.     There  is  a  slight  diminution  in  weight  immediately  after 

*  Lupiani  e  Lo  Monaco,  Arch.  Ital.  de  Biologie,  xxvn,  p.  340,  1897. 
t  Schaper,  Arch.  f.  Entwickdmigsmech.  xiv,  p.  35fi,  1902.     Cf.  Barfiirth,  Ver- 
suche  iiber  die  Verwandlung  der  Froschlarven,  Arch.  f.  mikr.  Anat.  xxix,  1887. 

6—2 


84 


THE   RATE   OF   GROWTH 


[CH. 


the  little  larva  frees  itself  from  the  egg ;  there  is  a  retardation  of 
growth  about  ten  days  later,  when  the  external  gills  disappear; 
and  finally,  the  complete  metamorphosis,  with  the  loss  of  the  tail, 


10 


15 


20 


25 


30 


„ 

e 

mgms 

1 

4000 

3000 

- 

/ 

r 

2000 

- 

/ 

\ 

1000 

- 

IV     / 

0/ 

1 

11 

III     / 

1 

'                        1                         ; 

35 


40 
days 


Fig.  13      Growth  in  weight  of  Silkworm.     (From  Ostwald,  after  Luciani 
and  Lo  Monaco.) 


the  growth  of  the  legs  and  the  cessation  of  branchial  respiration, 
is  accompanied  by  a  loss  of  weight  amounting  to  wellnigh  half 
the  weight  of  the  full-grown  larva. 


Ill] 


ITS  PERIODIC  RETARDATION 


85 


While  as  a  general  rule,  the  better  the  animals  be  fed  the 
quicker  they  grow  and  the  sooner  they  metamorphose,  Barfiirth 
has  pointed  out  the  curious  fact  that  a  short  spell  of  starvation, 
just  before  metamorphosis  is  due,  appears  to  hasten  the  change. 


Tngms. 

/ 

6000 

5000 

- 

/         "^ 
/          ^ 

/           ^ 

/                  o 

/              5 

/                    ^ 

4000 

/ 

3000 
2000 

- 1 

J2 
'5, 
a 

/ 

'■' 

1000 

-  1 

J! 

■^ 

y 

( 

60         70        80       90 
days 


0  10        20        30        40        50 

Fig.   14.     Growth  in  weight  of  Tadpole.     (From  Ostwald,  after  Schaper.) 


The  negative  growth,  or  actual  loss  of  bulk  and  weight  which 
often,  and  perhaps  always,  accompanies  metamorphosis,  is  well 
shewn  in  the  case  of  the  eel*.    The  contrast  of  size  is  great  between 

*  Joh.  Schmidt,  Contributions  to  the  Life-history  of  the  Eel,  Rapports  du  Conseil 
Intern,  pour  V exploration  de  la  Mer,  vol.  v,  pp.  137-274,  Copenhague,  1906. 


Fig,  15.     Development  of  Eel;   from  Leptocephalus  larvae  to  youno 
Elver.     (From  Ostwakl    after  Joh.  Schmidt.) 


CH.  Ill]  THE  KATE   OF   GROWTH  87 

the  flattened,  lancet-shaped  Leptocephalus  larva  and  the  httle 
black  cylindrical,  almost  thread-like  elver,  whose  magnitude  is 
less  than  that  of  the  Leptocephalus  in  every  dimension,  even,  at 
first,  in  length  (Fig.  15). 

From  the  higher  study  of  the  physiology  of  growth  we  learn 
that  such  fluctuations  as  we  have  described  are  but  special  inter- 
ruptions in  a  process  which  is  never  actually  continuous,  but  is 
perpetually  interrupted  in  a  rhythmic  manner*.  Hofmeister 
shewed,  for  instance,  that  the  growth  of  Spirogyra  proceeds  by 
fits  and  starts,  by  periods  of  activity  and  rest,  which  alternate 
with  one  another  at  intervals  of  so  many  minutes  (Fig.  16).     And 


20        40        60        80        100      120      140      160       180      200      220      240  260 

minutes 

Fig.   16.     Growth  in  length  of  Spirogyra.     (From  Ostwald,  after  Hofmeister.) 

Bose,  by  very  refined  methods  of  experiment,  has  shewn  that 
plant-growth  really  proceeds  by  tiny  and  perfectly  rhythmical 
pulsations  recurring  at  regular  intervals  of  a  few  seconds  of  time. 
Fig.  17  shews,  according  to  Bose's  observationsf,  the  growth  of 
a  crocus,  under  a  very  high  magnification.  The  stalk  grows  by 
little  jerks,  each  with  an  ampHtude  of  about  -002  mm.,  every 

*  That  the  metamorphoses  of  an  insect  are  but  phases  in  a  process  of 
growth,  was  firstly  clearly  recognised  by  Swammerdam,  Bihlia  Naturae,  1737, 
pp.  6,  579  etc 

t  From  Bose,  J.  C,  Plant  Eesponse,  London,  1906,  p.  417. 


88 


THE   RATE   OF   GROWTH 


[CH. 


twenty  seconds  or  so,  and  after  each  little  increment  there  is  a 
partial  recoil. 


Fig.  17.     Pulsations  of  growth  in  Crocus,  in  micro-millimetres. 
(After  Bose.) 


The  rate  of  growth  of  various  farts  or  organs*. 

The  differences  in  regard  to  rate  of  growth  between  various 
parts  or  organs  of  the  body,  internal  and  external,  can  be  amply 
illustrated  in  the  case  of  man,  and  also,  but  chiefly  in  regard  to 
external  form,  in  some  few  other  creaturesf.  It  is  obvious  that 
there  hes  herein  an  endless  field  for  the  mathematical  study  of 
correlation  and  of  variabihty,  but  with  this  aspect  of  the  case  we 
cannot  deal. 

In  the  accompanying  table,  I  shew,  from  some  of  Vierordt's 
data,  the  relative  weights,  at  various  ages,  compared  with  the 
weight  at  birth,  of  the  entire  body,  of  the  brain,  heart  and  hver ; 

*  This  phenomenon,  of  incrementum  inequale,  as  opposed  to  incrementum  in 
iiniversum,  was  most  carefully  studied  by  HaUer:  "Incrementum  inequale  multis 
modis  fit,  ut  aliae  partes  corporis  aliis  celerius  increscant.  Diximus  hepar  minus 
fieri,  majorem  pulmonem,  minimum  thymum.  etc."    (Elem.  \ui  (2),  p.  34). 

f  See  [inter  alia)  Fischel,  A.,  Variabilitat  und  Wachsthum  des  embryonalen 
Korpers,  Morphol.  Jahrb.  xxiv,  pp.  369-404,  18fi6.  Oppel,  VergUichung  des 
Entwickelungsgrades  der  Organe  zu  verschiedenen  Entwickelungszeiten  bei  Wirbel- 
thieren,  Jena,  1891.  Faucon,  A.,  Pesees  ef  Mensurations  fcetales  n  differents  ages 
de  la  grossesse.  (These.)  Paris,  1897.  Loisel,  G.,  Croissance  comparee  en  poids 
et  en  longueur  des  foetus  male  et  femelle  dans  I'espece  humaine,  C.  R.  Soc.  de 
Biologie,  Paris,  1903.  Jackson,  C.  M.,  Pre-natal  growth  of  the  human  body  and 
the  relative  growth  of  the  various  organs  and  parts.  Am.  J.  of  Anat.  ix,  1909; 
Post-natal  growth  and  variability  of  the  body  and  of  the  various  organs  in  the 
albino  rat,  ibid,  xv,  1913. 


Ill]  OF  PARTS   OR   ORGANS  89 

and  also  the  percentage  relation  which  each  of  these  organs  bears, 
at  the  several  ages,  to  the  weight  of  the  whole  body. 

Weight  of  Various  Organs,  compared  with  the  Total  Weight  of 
the  Human  Body  (male).  {After  Vierordt,  Anatorn.  Tabellen, 
pp.  38,  39.) 


Weight 

of  body'' 

in  kg. 

Relative 

weights  1 

A 

of 

Percentage  weights  compai-ed 
with  total  body-weights 

Age 

Body 

Brain 

Heart 

Liver 

Body 

Brain 

Heart 

Liver 

0 

31 

1 

1 

1 

1 

100 

12-29 

0-76 

4-57 

1 

9-0 

2-90 

2-48 

1-75 

2-35 

100 

10-50 

0-46 

3-70 

2 

11-0 

3-55 

2-69 

2-20 

3-02 

100 

9-32 

0-47 

3-89 

3 

12-5 

4-03 

2-91 

2-75 

3-42 

100 

8-86 

0-52 

3-88 

4 

14-0 

4-52 

3-49 

314 

4-15 

100 

9-50 

0-53 

4-20 

o 

15-9 

5-13 

3-32 

3-43 

3-80 

100 

7-94 

0-51 

3-39 

6 

17-8 

5-74 

3-57 

3-60 

4-34 

100 

7-63 

0-48 

3-45 

7 

19-7 

6-35 

S-54 

3-95 

4-86 

100 

6-84 

0-47 

3-49 

8 

21-6 

6-97 

3-62 

4-02 

4-59 

100 

6-38 

0-44 

3-01 

9 

23-5 

7-58 

3-74 

4-59 

4-95 

100 

6-06 

0-46 

2-99 

10 

25-2 

8-13 

3-70 

5-41 

5-90 

100 

5-59 

0-51 

3-32 

11 

27-0 

8-71 

3-57 

5-97 

6-14 

100 

5-04 

0-52 

3-22 

12 

29-0 

9-35 

3-78 

(4-13) 

6-21 

100 

4-88 

(0-34) 

3-03 

13 

331 

10-68 

3-90 

6-95 

7-31 

100 

4-49 

0-50 

3-13 

14 

37- 1 

11-97 

3-38 

9-16 

8-39 

100 

3-47 

0-58 

3-20 

15 

41-2 

13-29 

3-91 

8-45 

9-22 

100 

3-62 

0-48 

3-17 

16 

45-9 

14-81 

3-77 

9-76 

9-45 

100 

3-16 

0-51 

2-95 

17 

49-7 

16-03 

3-70 

10-63 

10-46 

100 

2-84 

0-51 

2-98 

18 

53-9 

17-39 

3-73 

10-33 

10-65 

100 

2-64 

0-46 

2-80 

19 

57-6 

18-58 

3-67 

11-42 

11-61 

100 

2-43 

0-51 

2-86 

20 

59-5 

19-19 

3-79 

12-94 

1101 

100 

2-43 

0-51 

2-62 

21 

61-2 

19-74 

3-71 

12-59 

11-48 

100 

2-31 

0-49 

2-66 

22 

62-9 

20-29 

3-54 

13-24 

11-82 

100 

2-14 

0-50 

2-66 

23 

(34-5 

20-81 

3-66 

12-42 

10-79 

100 

2-16 

0-46 

2-37 

24 

— 

— 

3-74 

13-09 

13-04 

100 

— 

— 

— 

25 

66-2 

21-36 

3-76 

12-74 

12-84 

100 

2-16 

0-46 

2-75 

* 

=  From  Quetelet. 

From  the  first  portion  of  the  table,  it  will  be  seen  that  none 
of  these  organs  by  any  means  keep  pace  with  the  body  as  a  whole 
in  regard  to  growth  in  weight;  in  other  words,  there  must  be 
some  other  part  of  the  fabric,  doubtless  the  muscles  and  the  bones, 
which  increase  more  rapidly  than  the  average  increase  of  the  body. 
Heart  and  liver  both  grow  nearly  at  the  same  rate,  and  by  the 


90 


THE  RATE   OF   GROWTH 


[CH. 


age  of  twenty-five  they  have  multiphed  their  weight  at  birth  by 
about  thirteen  times,  while  the  weight  of  the  entire  body  has  been 
multiphed  by  about  twenty-one ;  but  the  weight  of  the  brain  has 
meanwhile  been  multiphed  only  about  three  and  a  quarter  times. 
In  the  next  place,  we  see  the  very  remarkable  phenomenon  that 
the  brain,  growing  rapidly  till  the  child  is  about  four  years  old,  then 
grows  more  much  slowly  till  about  eight  or  nine  years  old,  and 
after  that  time  there  is  scarcely  any  further  perceptible  increase. 
These  phenomena  are  diagrammatically  illustrated  in  Fig.  18. 


0  5  10  15  20       ^ears    25 

Fig.  18.     Relative  growth  in  weight  (in  Man)  of  Brain,  Heart,  and 
whole  Body. 

Many  statistics  indicate  a  decrease  of  brain-weight  during  adult  life. 
Boas*  was  inclined  to  attribute  this  apparent  phenomenon  to  our  statistical 
methods,  and  to  hold  that  it  could  "  hardly  be  explained  in  any  other  way 
than  by  assuming  an  increased  death-rate  among  men  with  very  large  brains,, 
at  an  age  of  about  twenty  years."  But  Raymond  Pearl  has  shewn  that  there 
is  evidence  of  a  steady  and  very  gradual  decline  in  the  weight  of  the  brain 
with  advancing  age,  beginning  at  or  before  the  twentieth  year,  and  con- 
tinuing throughout  adult  lifef. 

*  I.e.  p.  1542. 

t  Variation  and  Correlation  in  Brain-weight,  Biometrika,  iv,  pp.  13-104,  1905. 


Ill]  OF   PAKTS   OR   ORGANS  91 

The  second  part  of  the  table  shews  the  steadily  decreasing 
weights  of  the  organs  in  question  as  compared  with  the  body; 
the  brain  falling  from  over  12  per  cent,  at  birth  to  little  over 
2  per  cent,  at  five  and  twenty;  the  heart  from  -75  to  -46  per 
cent. ;  and  the  hver  from  4-57  to  2-75  per  cent,  of  the  whole 
bodily  weight. 

It  is  plain,  then,  that  there  is  no  simple  and  direct  relation, 
holding  good  throughout  life,  between  the  size  of  the  body  as  a 
whole  and  that  of  the  organs  we  have  just  discussed;  and  the 
changing  ratio  of  magnitude  is  especially  marked  in  the  case  of 
the  brain,  which,  as  we  have  just  seen,  constitutes  about  one-eighth 
of  the  whole  bodily  weight  at  birth,  and  but  one-fiftieth  at  five 
and  twenty.  The  same  change  of  ratio  is  observed  in  other 
animals,  in  equal  or  even  greater  degree.  For  instance.  Max 
Weber*  tells  us  that  in  the  Hon,  at  five  weeks,  four  months, 
eleven  months,  and  lastly  when  full-grown,  the  brain-weight 
represents  the  following  fractions  of  the  weight  of  the  whole 
body,  viz.  1/18,  1/80,  1/184,  and  1/546.  And  KelHcott  has,  in 
hke  manner,  shewn  that  in  the  dogfish,  while  some  organs  (e.g. 
rectal  gland,  pancreas,  etc.)  increase  steadily  and  very  nearly 
proportionately  to  the  body  as  a  whole,  the  brain,  and  some  other 
organs  also,  grow  in  a  diminishing  ratio,  which  is  capable  of 
representation,  approximately,  by  a  logarithmic  curve  f. 

But  if  we  confine  ourselves  to  the  adult,  then,  as  Raymond 
Pearl  has  shewn  in  the  case  of  man,  the  relation  of  brain-weight 
to  age,  to  stature,  or  to  weight,  becomes  a  comparatively  simple 
one,  and  rnay  be  sensibly  expressed  by  a  straight  line,  or  simple 
equation. 

Thus,  if  W  be  the  brain-weight  (in  grammes),  and  A  be  the 
age,  or  S  the  stature,  of  the  individual,  then  (in  the  case  of  Swedish 
males)  the  following  simple  equations  suffice  to  give  the  required 
ratios : 

W  =  1487-8  -  1-94^  =  915-06  -f  2-86  iS. 

*  Die  Sdugethiere,  p.  117. 

t  Amer.  J.  of  Anatomy,  vm,  pp.  319-353,  1908.  Donaldson  {Journ.  Camp. 
Neur.  and  Psychol,  xvm,  pp.  345-392,  1908)  also  gives  a  logarithmic  formula  for 
brain-weight  (y)  as  compared  with  body-weight  (x),  which  in  the  case  of  the  white 
rat  is  J,  =  -554  -^  -569  log  (a;— 8-7),  and  the  agreement  is  very  close.  But  the 
formula  i&  admittedly  empinca  and  as  Raymond  Pearl  says  (Amer.  Nat.  1909, 
p.  303),  "  no  ulterior  biological  significance  is  to  be  attached  to  it." 


92 


THE   RATE   OF   GROWTH 


[CH. 


These  equations  are  applicable  to  ages  between  fifteen  and  eighty ; 
if  we  take  narrower  limits,  say  between  fifteen  and  fifty,  we  can  get 
a  closer  agreement  by  using  somewhat  altered  constants.  In  the 
two  sexes,  and  in  different  races,  these  empirical  constants  will  be 
greatly  changed* .  Donaldson  has  further  shewn  that  the  correla- 
tion between  brain-weight  and  body-weight  is  very  much  closer 
in  the  rat  than  in  man"]". 

The  falling  ratio  of  weight  of  brain  to  body  with  increase  of  size  or  age 
finds  its  parallel  in  comparative  anatomy,  in  the  general  law  that  the  larger 
the  animal  the  less  is  the  relative  weight  of  the  brain. 


Weight  of 

Weight  of 

entire  animal 

brain 

gms. 

gms. 

Ratio 

Marmoset 

335 

12-5 

26 

Spider  monkey 

1845 

126 

15 

FeUs  minuta    . . . 

1234 

23-6 

56 

F.  domestica    ... 

3300 

31 

107 

Leopard 

27,700 

164 

168 

Lion 

119,500 

219 

546 

Elephant 

3,048,000 

5430 

560 

Whale  (Globiocephak 

is)          1,000,000 

2511 

400 

For  much  information  on  this  subject,  see  Dubois,  "  Abhangigkeit  des 
Hirngewichtes von  der Korpergrosse  bei  denSsiUgethieren,'''  Arch.  f.  Anthropol. 
XXV,  1897.  Dubois  has  attempted,  but  I  think  with  very  doubtful  success, 
to  equate  the  weight  of  the  brain  with  that  of  the  animal.  We  may  do  this, 
in  a  very  simple  way,  by  representing  the  weight  of  the  body  as  a  power  of 
that  of  the  brain ;  thus,  in  the  above  table  of  the  weights  of  brain  and  body 
in  four  species  of  cat,  if  we  call  W  the  weight  of  the  body  (in  grammes),  and 
w  the  weight  of  the  brain,  then  if  in  all  four  cases  we  express  the  ratio  by 
W  —  w",  we  find  that  n  is  almost  constant,  and  differs  little  from  2-24  in  all 
four  species:  the  values  being  respectively,  in  the  order  of  the  table  2-36, 
2-24,  2-18,  and  2-17.  But  this  evidently  amounts  to  no  more  than  an 
empirical  rule;  for  we  can  easily  see  that  it  depends  on  the  particular  scale 
which  we  have  used,  and  that  if  the  weights  had  been  taken,  for  instance, 
in  kilogrammes  or  in  milligrammes,  the  agreement  or  coincidence  would  not 
have  occurred  J. 

*   Biometrika,  iv,  pp.  13-104,  1904. 

t  Donaldson,  H.  H.,  A  Comparison  of  the  White  Rat  with  Man  in  respect  to 
the  Growth  of  the  entire  Body,  Boas  Memorial  Vol.,  New  York,  1906,  pp.  5-26. 

t  Besides  many  papers  quoted  by  Dubois  on  the  growth  and  weight  of  the 
brain,  and  numerous  papers  in  Biometrika,  see  also  the  following:  Ziehen,  Th., 
Das  Gehirn :  Massverhaltnisse,  in  Bardeleben's  Handh.  der  Anat.  des  Menschen, 
IV,  pp.  353-386,  1899.  Spitzka,  E.  A.,  Brain-weight  of  Animals  with  special 
reference  to  the  Weight  of  the  Brain  in  the  Macaque  Monkey,  J.  Comp.  Neurol. 


Ill] 


OF  PARTS   OR   ORGANS 

The  Length  of  the  Head  in  Man  at  various  Ages. 
{After  Quetelet,  f.  207.) 

Men  Women 


93 


Age 

Total  height 

Head 

Ratio 

Height 

Head* 

Ratio 

m. 

m. 

m. 

m. 

Birth 

0-500 

0111 

4-50 

0-494 

0-111 

4-45 

1  year 

0-698 

0-154 

4-53 

0-690 

0-154 

4-48 

2  years 

0-791 

0-173 

4-57 

0-781 

0-172 

4-54 

3      „ 

0-864 

0-182 

4-74 

0-854 

0-180 

4-74 

5      „ 

0-987 

0-192 

514 

0-974 

0-188 

5-18 

10      „ 

1-273 

0-205 

6-21 

1-249 

0-201 

6-21 

15      „ 

1-513 

0-215 

7-04 

1-488 

0-213 

6-99 

20      „ 

1-669 

0-227 

7-35 

1-574 

0-220 

7-15 

30      „ 

1-686 

0-228 

7-39 

1-580 

0-221 

7-15 

40      „ 

1-686 

0-228 

7-39 

1-580 

0-221 

715 

*  A  smooth  curve,  very  similar  to  this,  for  the  growth  in  "auricular  height" 
of  the  girl's  head,  is  given  by  Pearson,  in  Biometrika,  iii,  p.  141,  1904. 

As  regards  external  form,  very  similar  differences  exist,  which 
however  we  must  express  in  terms  not  of  weight  but  of  length. 
Thus  the  annexed  table  shews  the  changing  ratios  of  the  vertical 
length  of  the  head  to  the  entire  stature ;  and  while  this  ratio 
constantly  diminishes,  it  will  be  seen  that  the  rate  of  change  is 
greatest  (or  the  coefficient  of  acceleration  highest)  between  the 
ages  of  about  two  and  five  years. 

In  one  of  Quetelet's  tables  {supra,  p.  63),  he  gives  measure- 
ments of  the  total  span  of  the  outstretched  arms  in  man,  from 
year  to  year,  compared  with  the  vertical  stature.  The  two 
measurements  are  so  nearly  identical  in  actual  magnitude  that  a 
direct  comparison  by  means  of  curves  becomes  unsatisfactory; 
but  I  have  reduced  Quetelet's  data  to  percentages,  and  it  will  be 
seen  from  Fig.  19  that  the  percentage  proportion  of  span  to 
height  undergoes  a  remarkable  and  steady  change  from  birth  to 
the  age  of  twenty  years ;  the  man  grows  more  rapidly  in  stretch 
of  arms  than  he  does  in  height,  and  the  span  which  was  less  than 

xm,  pp.  &-17,  1903.  Warneke,  P.,  Mitteilimg  neuer  Gehirn  imd  Korperge- 
wichtsbestimmungen  bei  Saugern,  neb.st  Zusammenstellung  der  gesammten  bisher 
beobachteten  absoluten  imd  relativen  Gehirngewichte  bei  den  verschiedenen 
Species,  J.  f.  Psychol,  u.  Neurol,  xm,  pp.  355—403,  1909.  Donaldson,  H.  H.,  On 
'  the  regular  seasonal  Changes  in  the  relative  Weight  of  the  Central  Nervous  System 
of'the  Leopard  Frog,  Journ.  of  Morph.  xxii,  pp.  663-694,  1911. 


94 


THE   RATE   OF   GROWTH 


[CH. 


the  stature  at  birth  by  about  1  per  cent,  exceeds  it  at  the  age  of 
twenty  by  about  4  per  cent.  After  the  age  of  twenty,  Quetelet's 
data  are  few  and  irregular,  but  it  is  clear  that  the  span  goes  on 
for  a  long  while  increasing  in  proportion  to  the  stature.  How 
far  the  phenomenon  is  due  to  actual  growth  of  the  arms  and 
how  far  to  the  increasing  breadth  of  the  chest  is  not  yet  ■ 
ascertained. 


Fig.  19.     Ratio  of  stature  in  Man,  to  span  of  outstretched  arms. 
(From  Quetelet's  data.) 


The  differences  of  rate  of  growth  in  different  parts  of  the  body 
are  very  simply  brought  out  by  the  following  table,  which  shews 
the  relative  growth  of  certain  parts  and  organs  of  a  young  trout, 
at  intervals  of  a  few  days  during  the  period  of  most  rapid  develop- 
ment. It  would  not  be  difficult,  from  a  picture  of  the  little 
trout  at  any  one  of  these  stages,  to  draw  its  approximate  form 
at  any  other,  by  the  help  of  the  numerical  data  here  set 
forth*. 


*  Cf.    Jenkinson,    Growth,    Variability    and    Correlation    in    Young    Trout, 
Biometrika,  viii.  pp.  444-455,  1912. 


Ill]  OF   PARTS   OR   ORGANS  95 

Trout  {Scdnio  fario)  :   pwportionate  groivth  of  various  organs. 
{From  Jenkinson's  data.) 


Days 

Total 

1st 

Ventral 

2nd 

Breadth 

old 

length 

Eye 

Head 

dorsal 

fin 

dorsal 

Tail-fin 

of  tail 

49 

100 

100 

100 

100 

100 

100 

100 

100 

63 

129-9 

129-4 

148-3 

148-6 

148-5 

108-4 

173-8 

155-9 

11 

154-9 

147-3 

189-2 

(203-6) 

(193-6) 

139-2 

257-9 

220-4 

92 

173-4 

179-4 

220-0 

(193-2) 

(182-1) 

154-5 

307-6 

272-2 

106       194-6       192-5       242-5       173-2       165-3       173-4       337-3      287-7 

While  it  is  inequality  of  growth  in  different  directions  that  we 
can  most  easily  comprehend  as  a  phenomenon  leading  to  gradual 
change  of  outward  form,  we  shall  see  in  another  chapter*  that 
differences  of  rate  at  different  parts  of  a  longitudinal  system, 
though  always  in  the  same  direction,  also  lead  to  very  notable 
and  regular  transformations.  Of  this  phenomenon,  the  difference 
in  rate  of  longitudinal  growth  between  head  and  body  is  a  simple 
case,  and  the  difference  which  accompanies  and  results  from  it  in 
the  bodily  form  of  the  child  and  the  man  is  easy  to  see.  A  like 
phenomenon  has  been  studied  in  much  greater  detail  in  the  case 
of  plants,  by  Sachs  and  certain  other  botanists,  after  a  method 
in  use  by  Stephen  Hales  a  hundred  and  fifty  years  before  f. 

On  the  growing  root  of  a  bean,  ten  narrow  zones  were  marked 
off,  starting  from  the  apex,  each  zone  a  milhmetre  in  breadth. 
After  twenty-four  hours'  growth,  at  a  certain  constant  tempera- 
ture, the  whole  marked  portion  had  grown  from  10  mm.  to  33  mm. 
in  length ;  but  the  individual  zones  had  grown  at  very  unequal 
rates,  as  shewn  in  the  annexed  tablet. 


Zone 

Increment 

mm. 

Apex 

1-5 

2nd 

5-8 

3rd 

8-2 

4th 

3-5 

5th 

1-6 

Zone 

Increment 

2nm. 

6th 

1-3 

7th 

0-5 

8th 

0-3 

9th 

0-2 

10th 

0-1 

*  Cf.   chap,   xvii,   p.   739. 

t  "  ...I  marked  in  the  same  manner  as  the  Vine,  young  Honeysuckle  shoots, 
etc. . . . ;  and  I  found  in  them  all  a  gradual  scale  of  unequal  extensions,  those  parts 
extending  most  which  were  tenderest,"  Vegetable  Staiicks,  Exp.  cxxili. 

t  From  Sachs,  Textbook  of  Botany,   1882,  p.  820. 


96 


THE   RATE   OF   GROWTH 


[CH. 


The  several  values  in  this  table  he  very  nearly  (as  we  see  by 
Fig.  20)  in  a  smooth  curve;  in  other  words  a  definite  law,  or 
principle  of  continuity,  connects  the  rates  of  growth  at  successive 
points  along  the  growing  axis  of  the  root.  Moreover  this  curve, 
in  its  general  features,  is  singularly  hke  those  acceleration-curves 
which  we  have  already  studied,  in  which  we  plotted  the  rate  of 
growth  against  successive  intervals  of  time,  as  here  we  have 
plotted  it  against  successive  spatial  intervals  of  an  actual  growing 


9 

mm. 

III' 

II                       1                      ! 

8 

A' 

- 

7 

/  \ 

- 

6 

/  \ 

- 

5 

-    /    \ 

- 

4 

-    /     \ . 

- 

3 

-  /       \ 

- 

2 

/        ^ 

1 

/    1     1     1     1 

1               1               ■              '        ^Tr 

0  123456789      10 

Zones 

Fig.  20.     Rate  of  growth  in  successive  zones  near  the  tip  of  the  bean-root. 


structure.  If  we  suppose  for  a  moment  that  the  velocities  of 
growth  had  been  transverse  to  the  axis,  instead  of,  as  in  this  case, 
longitudinal  and  parallel  with  it,  it  is  obvious  that  these  same 
velocities  would  have  given  us  a  leaf-shaped  structure,  of  which 
our  curve  in  Fig.  20  (if  drawn  to  a  suitable  scale)  would  represent 
the  actual  outhne  on  either  side  of  the  median  axis;  or,  again, 
if  growth  had  been  not  confined  to  one  plane  but  symmetrical 
about  the  axis,  we  should  have  had  a  sort  of  turnip-shaped  root, 


Ill]  OF   PARTS   OR   ORGANS  97 

having  the  form  of  a  surface  of  revolution  generated  by  the  same 
curve.  This  then  is  a  simple  and  not  unimportant  illustration  of 
the  direct  and  easy  passage  from  velocity  to  form. 

A  kindred  problem  occurs  when,  instead  of  "' zones "'  artificially  marked  out 
in  a  stem,  we  deal  with  the  rates  of  growth  in  successive  actual  "internodes"  ; 
and  an  interesting  variation  of  this  problem  occurs  when  we  consider,  not  the 
actual  growth  of  the  internodes,  but  the  varying  number  of  leaves  which  they 
successively  produce.  Where  we  have  whorls  of  leaves  at  each  node,  as  in 
Equisetum  and  in  many  water-weeds,  then  the  problem  presents  itself  in  a 
simple  form,  and  in  one  such  case,  namely  in  Ceratophyllum,  it  has  been 
carefully  investigated  by  Mr  Raymond  Pearl*. 

It  is  found  that  the  mean  number  of  leaves  per  whorl  increases  with  each 
successive  whorl;  but  that  the  rate  of  increment  diminishes  from  whorl  to 
whorl,  as  we  ascend  the  axis.  In  other  words,  the  increase  in  the  number  of 
leaves  per  whorl  follows  a  logarithmic  ratio;  and  if  y  be  the  mean  number  of 
leaves  per  whorl,  and  x  the  successional  number  of  the  whorl  from  the  root 
or  main  stem  upwards,  then 

y  —  A  +  C  log  {x  -  a), 

where  A,  C,  and  a  are  certain  specific  constants,  varying  with  the  part  of  the 
plant  which  we  happen  to  be  considering.  On  the  main  stem,  the  rate  of 
change  in  the  number  of  leaves  per  whorl  is  very  slow;  when  we  come  to  the 
small  twigs,  or  "tertiary  branches,"  it  has  become  rapid,  as  we  see  from  the 
following  abbreviated  table : 

Number  of  leaves  per  whorl  on  the  tertiary  branches  of  Ceratophyllum. 

Position  of  whorl       ...  1  2  3  4  5  6 

Mean  number  of  leaves       6-55       8-07       9-00       9-20       9-75       10-00 
Increment       —        1-52         -93         -20       (-55)        (-25) 


We  have  seen  that  a  slow  but  definite  change  of  form  is  a 
common  accompaniment  of  increasing  age,  and  is  brought  about 
as  the  simple  and  natural  result  of  an  altered  ratio  between  the 
rates  of  growth  in  different  dimensions :  or  rather  by  the  pro- 
gressive change  necessarily  brought  about  by  the  difference  in 
their  accelerations.  There  are  many  cases  however  in  which 
the  change  is  all  but  imperceptible  to  ordinary  measurement, 
and  many  others  in  which  some  one  dimension  is  easily  measured, 
but   others    are    hard  to  measure  with  corresponding  accuracy. 

*  Variation  and  Differentiation  in  Ceratophyllum,  Carneqie  Inst.  Publica- 
tions, No.  58,  Washington,  1907. 


98  THE   RATE   OF   GROWTH  [ch. 

For  instance,  in  any  ordinary  fish,  such  as  a  plaice  or  a  haddock, 
the  length  is  not  difficult  to  measure,  but  measurements  of 
breadth  or  depth  are  very  much  more  uncertain.  In  cases  such 
as  these,  while  it  remains  difficult  to  define  the  precise  nature  of 
the  change  of  form,  it  is  easy  to  shew  that  such  a  change  is 
taking  place  if  we  make  use  of  that  ratio  of  length  to  weight 
which  we  have  spoken  of  in  the  preceding  chapter.  Assuming,  as 
we  may  fairly  do,  that  weight  is  directly  proportional  to  bulk  or 
volume,  we  may  express  this  relation  in  the  form  WJL^  =  k,  where 
k  is  a  constant,  to  be  determined  for  each  particular  case.  {W 
and  L  are  expressed  in  grammes  and  centimetres,  and  it  is  usual 
to  multiply  the  result  by  some  figure,  such  as  1000,  so  as  to  give 
the  constant  k  a  value  near  to  unity.) 

Plaice  caught  in  a  certain  area,  March,  1907.  Variation  of  k  (the 
weight-length  coefficient)  with  size.  {Data  taken  from  the 
Department  of  Agriculture  and  Fislieries'  Plaice- Report, 
vol.  I,  f.  107,  1908.) 

Size  in  cm.   Weight  in  gm.  WjL^  x  10,000   WjL^  (smoothed) 

23  113  92-8           — 

24  128  92-6  94-3 

25  152  97-3  96-1 

26  173  98-4  97-9 

27  193  98-1  990 

28  221  100-6  100-4 

29  250  102-5  101-2 

30  271  100-4  101-2 

31  300  100-7  100-4 

32  328  100-1  99-8 

33  354  98-5  98-8 

34  384  97-7  98-0 

35  419  97-7  97-6 

36  454  97-3  96-7 

37  492  95-2  96-3 

38  529  96-4  95-6 

39  564  95-1  95-0 

40  614  95-9  95-0 

41  647  93-9  93-8 

42  679  91-6  92-5 

43  732  92-1  92-5 

44  800  93-9  94-0 

45  875  96-0           — 


Ill] 


THE   WEIGHT-LENGTH   COEFFICIENT 


99 


Now  while  this  k  may  be  spoken  of  as  a  "constant,"  having 
a  certain  mean  value  specific  to  each  species  of  organism,  and 
depending  on  the  form  of  the  organism,  any  change  to  which  it 
may  be  subject  will  be  a  very  delicate  index  of  progressive  changes 
of  form ;  for  we  know  that  our  measurements  of  length  are,  on 
the  average,  very  accurate,  and  weighing  is  a  still  more  dehcate 
method  of  comparison  than  any  linear  measurement. 

Thus,  in  the  case  of  plaice,  when  we  deal  with  the  mean  values 
for  a  large  number  of  specimens,  and  when  we  are  careful  to  deal 
only  with  such  as  are  caught  in  a  particular  locality  and  at  a  par- 
ticular time,  we  see  that  k  is  by  no  means  constant,  but  steadily 
increases  to  a  maximum,  and  afterwards  slowly  declines  with  the 


23    25      27      29       31       33      35       37      39      41       43cms. 
Fig.  21.     Changes  in  the  weight-length  ratio  of  Plaice,  with  increasing  size. 

increasing  size  of  the  fish  (Fig.  21).  To  begin  with,  therefore,  the 
weight  is  increasing  more  rapidly  than  the  cube  of  the  length,  and 
it  follows  that  the  length  itself  is  increasing  less  rapidly  than  some 
other  linear  dimension ;  while  in  later  fife  this  condition  is  reversed. 
The  maximum  is  reached  when  the  length  of  the  fish  is  somewhere 
near  to  30  cm.,  and  it  is  tempting  to  suppose  that  with  this  "point 
of  inflection"  there  is  associated  some  well-marked  epoch  in  the 
fish's  life.  As  a  matter  of  fact,  the  size  of  30  cm.  is  approximately 
that  at  which  sexual  maturity  may  be  said  to  begin,  or  is  at  least 
near  enough  to  suggest  a  close  connection  between  the  two 
phenomena.      The  first  step  tov^ards  further  investigation  of  the 

7—2 


100 


THE   RATE   OF   GROWTH 


[CH. 


apparent  coincidence  would  be  to  determine  the  coefficient  k  of 
the  two  sexes  separately,  and  to  discover  whether  or  not  the  point 
of  inflection  is  reached  (or  sexual  maturity  is  reached)  at  a  smaller 
size  in  the  male  than  in  the  female  plaice ;  but  the  material  for 
this  investigation  is  at  present  scanty. 

A  still  more  curious  and  more  unexpected  result  appears  when 
we  compare  the  values  of  k  for  the  same  fish  at  different  seasons  of 
the  year*.  When  for  simphcity's  sake  (as  in  the  accompanying 
table  and  Fig.  22)  we  restrict  ourselves  to  fish  of  one  particular 


Fig. 


JFMAMJ       JASONDJ 
22.     Periodic  annual  change  in  the  weight-length  ratio  of  Plaice. 


size,  it  is  not  necessary  to  determine  the  value  of  k,  because  a 
change  in  the  ratio  of  length  to  weight  is  obvious  enough ;  but 
when  we  have  small  numbers,  and  various  sizes,  to  deal  with, 
the  determination  of  k  may  help  us  very  much.  It  will  be  seen, 
then,  that  in  the  case  of  plaice  the  ratio  of  weight  to  length 
exhibits  a  regular  periodic  variation  with  the  course  of  the  seasons. 

*  Cf.  Lammel,  Ueber  periodische  Variationen  in  Organismen,  Biol.  Centralhl. 
xxn,  pp.  368-376,  1903. 


Ill]  THE   WEIGHT-LENGTH   COEFFICIENT  101 

RelMion  of  Weight  to  Length  in  Plaice  of  55  cm.  long,  from  Month 
to  Month.  {Data  taken  from  the  Department  of  Agriculture 
and  Fisheries'  Plaice- Report,  vol.  ii,  p.  92,  1909.) 


Average  weight 

in  grammes 

W/L^  X  100 

W/L^  (smoothed) 

Jan. 

2039 

1-226 

1-157 

Feb. 

1735 

1043 

1-080 

March 

1616 

0-97t 

0-989 

April 

1585 

0-953 

0-967 

May 

1624 

0-976 

0-985 

June 

1707 

1-026 

1-005 

July 

1686 

1-013 

1-037 

August 

1783 

1-072 

1-042 

Sept. 

1733 

1-042 

1-111 

Oct. 

2029 

1-220 

1-160 

Nov. 

2026 

1-218 

1-213 

Dec. 

1998 

1-201 

1-215 

With  unchanging  length,  the  weight  and  therefore  the  bulk  of  the 
fish  falls  off  from  about  November  to  March  or  April,  and  again 
between  May  or  June  and  November  the  bulk  and  weight  are 
gradually  restored.  The  explanation  is  simple,  and  depends 
wholly  on  the  process  of  spawning,  and  on  the  subsequent  building 
up  again  of  the  tissues  and  the  reproductive  organs.  It  follows 
that,  by  this  method,  without  ever  seeing  a  fish  spawn,  and  without 
ever  dissecting  one  to  see  the  state  of  its  reproductive  system,  we 
can  ascertain  its  spawning  season,  and  determine  the  beginning 
and  end  thereof,  with  great  accuracy. 

As  a  final  illustration  of  the  rate  of  growth,  and  of  unequal 
growth  in  various  directions,  I  give  the  following  table  of  data 
regarding  the  ox,  extending  over  the  first  three  years,  or  nearly 
so,  of  the  animal's  life.  The  observed  data  are  (1)  the  weight  of 
the  animal,  month  by  month,  (2)  the  length  of  the  back,  from  the 
occiput  to  the  root  of  the  tail,  and  (3)  the  height  to  the  withers. 
To  these  data  I  have  added  (1)  the  ratio  of  length  to  height, 
(2)  the  coefficient  (k)  expressing  the  ratio  of  weight  to  the  cube  of 
the  length,  and  (3)  a  similar  coefficient  {k')  for  the  height  of  the 
animal.  It  will  be  seen  that,  while  all  these  ratios  tend  to  alter 
continuously,  shewing  that  the  animal's  form  is  steadily  altering 
as  it  approaches  maturity,  the  ratio  between  length  and  weight 


102 


THE   RATE   OF   GROWTH 


[CH. 


changes  comparatively  little.  The  simple  ratio  between  length 
and  height  increases  considerably,  as  indeed  we  should  expect; 
for  we  know  that  in  all  Ungulate  animals  the  legs  are  remarkably 


Relations  between  the  Weight  and  certain  Linear  Dimensions 
of  the  Ox.     {Data  from  Przihrayn,  after  Cornevin*.) 


Age  in 
months 

If,  wt. 
in  kg. 

L,  length 
of  back 

H,  height 

L/H 

xlO 

/■'  =  W/H 

xlO 

0 

37 

•78 

•70 

114 

•779 

1-079 

1 

55-3 

•94  . 

-77 

221 

•665 

1-210 

2 

86-3 

1-09 

-85 

282 

-666 

1-406 

3 

121-3 

1-207 

•94 

284 

-690 

1-460 

4 

150-3 

i-314 

•95 

383 

•662 

1^754 

5 

179-3 

1-404 

1-040 

350 

•649 

1-600 

6 

210-3 

1-484 

1-087 

365 

-644 

1-638 

7    . 

247-3 

1-524 

1-122 

358 

-699 

1-751 

8 

267-3 

1-581 

1-147 

378 

•677 

1-791 

9 

282-8 

1-621 

1-162 

395 

•664 

1-802 

10 

303-7 

1-651 

1-192 

385 

•675 

1-793 

11 

327-7 

1-694 

1-215 

394 

•674 

1-794 

12 

350-7 

1-740 

1-238 

405 

•666 

1-849 

13 

374-7 

1-765 

1-254 

407 

•682 

1-900 

14 

391-3 

1-785 

1-264 

412 

-688 

1-938 

15 

405-9 

1-804 

1-270 

420 

-692 

1-982 

16 

417-9 

1-814 

1-280 

417 

•700 

2-092 

17 

423-9 

1-832 

1-290 

420 

•689 

1-974 

18 

423-9 

1-859 

1-297 

433 

•660 

1-943 

19 

427-9 

1-875 

1-307 

435 

•649 

1-916 

20 

437-9 

1-884 

1-311 

437 

•655 

1-944 

21 

447-9 

1-893 

1-321 

433 

-661 

1-943 

22 

464-4 

1-901 

1-333 

426 

-676 

1-960 

23 

480-9 

1-909 

1-345 

419 

•691 

1-977 

24 

500-9 

1-914 

1-352 

416 

•714 

2-027 

25 

520-9 

1-919 

1-359 

412 

-737 

2-075 

26 

534-1 

1-924 

1-361 

414 

•750 

2-119 

27 

547-3 

1-929 

1-363 

415 

•762 

2-162 

28 

554-5 

1-929 

1-363 

415 

-772 

2-190 

29 

561-7 

1-929 

1-363 

415 

-782 

2-218 

30 

586-2 

1-949 

1-383 

409 

•792 

2-216 

31 

610-7 

1-969 

1-403 

403 

-800 

2-211 

32 

625-7 

1-983 

1-420 

396 

-803 

2-186 

33 

640-7 

1-997 

1-437 

390 

•805 

2-159 

34 

655-7 

2-011 

1-454 

383 

-806 

2- 133 

*  Cornevin,  Ch.,  fitudes  sur  la  croissance,  Arch,  de  Physiol,  norm,  et  pathol. 
(5),  IV,  p.  477,  1892. 


Ill]  THE   WEIGHT-LENGTH   COEFFICIENT  103 

long  ;  t  birth  in  comparison  with  other  dimensions  of  the  body. 
It  is  somewhat  curious,  however,  that  this  ratio  seems  to  fall  off 
a  little  in  the  third  year  of  growth,  the  animal  continuing  to  grow 
in  height  to  a  marked  degree  after  growth  in  length  has  become 
very  slow.  The  ratio  between  height  and  weight  is  by  much  the 
most  variable  of  our  three  ratios ;  the  coefficient  W jH^  steadily 
increases,  and  is  more  than  twice  as  great  at  three  years  old  as 
it  was  at  birth.  This  illustrates  the  important,  but  obvious  fact, 
that  the  coefficient  k  is  most  variable  in  the  case  of  that 
dimension  which  grows  most  uniformly,  that  is  to  say  most  nearly 
in  proportion  to  the  general  bulk  of  the  animal.  In  short,  the 
successive  values  of  k,  as  determined  (at  successive  epochs)  for 
one  dimension,  are  a  measure  of  the  variability  of  the  others. 

From  the  whole  of  the  foregoing  discussion  we  see  that  a  certain 
definite  rate  of  growth  is  a  characteristic  or  specific  phenomenon, 
deep-seated  in  the  physiology  of  the  organism;  and  that  a  very 
large  part  of  the  specific  morphology  of  the  organism  depends  upon 
the  fact  that  there  is  not  only  an  average,  or  aggregate,  rate  of 
growth  common  to  the  whole,  but  also  a  variation  of  rate  in 
different  parts  of  the  organism,  tending  towards  a  specific  rate 
characteristic  of  each  different  part  or  organ.  The  smallest  change 
in  the  relative  magnitudes  of  these  partial  or  localised  velocities 
of  growth  will  be  soon  manifested  in  more  and  more  striking 
differences  of  form.  This  is  as  much  as  to  say  that  the  time- 
element,  which  is  implicit  in  the  idea  of  growth,  can  never  (or 
very  seldom)  be  wholly  neglected  in  our  consideration  of  form*. 
It  is  scarcely  necessary  to  enlarge  here  upon  our  statement,  for 
not  only  is  the  truth  of  it  self-evident,  but  it  will  find  illustration 
again  and  again  throughout  this  book.  Nevertheless,  let  us  go 
out  of  our  way  for  a  moment  to  consider  it  in  reference  to  a 
particular  case,  and  to  enquire  whether  it  helps  to  remove  any  of 
the  difficulties  which  that  case  appears  to  present. 

*  Herein  lies  the  easy  answer  to  a  contention  fiequently  raised  by  Bergson, 
and  to  which  he  asciibes  great  importance,  that  "a  mere  variation  of  size  is  one 
thing,  and  a  change  of  form  is  another."  Thus  he  considers  "a  change  in  the 
form  of  leaves"  to  constitute  "a  profound  morphological  difference."  Creative 
Evolution,  p.  71. 


104 


THE   RATE   OF   GROWTH 


[CH. 


'  In  a  very  well-known  paper,  Bateson  shewed  that,  among  a 
large  number  of  earwigs,  collected  in  a  particular  locality,  the 
males  fell  into  two  groups,  characterised  by  large  or  by  small 
tail-forceps,  with  very  few  instances  of  intermediate  magnitude. 
This  distribution  into  two  groups,  according  to  magnitude,  is 
illustrated  in  the  accompanying  diagram  (Fig.  23) ;  and  the 
phenomenon  was  described,  and  has  been  often  quoted,  as  one 
of  dimorphism,  or  discontinuous  variation.  In  this  diagram  the 
time-element  does  not  appear ;  but  it  is  certain,  and  evident,  that 
it  lies  close  behind.     Suppose  we  take  some  organism  which  is 


T3     100 


Length  of  tail-forceps,  in  mm. 

Fig.  23.     Variability  of  length  of  tail-forceps  in  a  sample  of  Earwigs. 
(After  Bateson,  P.  Z.  S.   1892,  p.  588.) 


born  not  at  all  times  of  the  year  (as  man  is)  but  at  some  one 
particular  season  (for  instance  a  fish),  then  any  random  sample 
will  consist  of  individuals  whose  ages,  and  therefore  whose  magni- 
tudes, will  form  a  discontinuous  series ;  and  by  plotting  these 
magnitudes  on  a  curve  in  relation  to  the  number  of  individuals 
of  each  particular  magnitude,  we  obtain  a  curve  such  as  that 
shewn  in  Fig.  24,  the  first  practical  use  of  which  is  to  enable  us 
to  analyse  our  sample  into  its  constituent  "age-groups,"  or  in 
•other  words  to  determine  approximately  the  age,  or  ages  of  the 
fish.  And  if,  instead  of  measuring  the  whole  length  of  our  fish, 
we  had  confined  ourselves  to  particular  parts,  such  as  head,  or 


Ill 


A   CASE   OF   DIMORPHISM 


105 


tail  or  fin,  we  should  have  obtained  discontinuous  curves  of 
distribution,  precisely  analogous  to  those  for  the  entire  animal. 
Now  we  know  that  the  differences  with  which  Bateson  was  dealing 
were  entirely  a  question  of  magnitude,  and  we  cannot  help  seeing 
that  the  discontinuous  distributions  of  magnitude  represented  by 
his  earwigs'  tails  are  just  such  as  are  illustrated  by  the  magnitudes 
of  the  older  and  younger  fish ;  we  may  indeed  go  so  far  as  to  say 
that  the  curves  are  precisely  comparable,  for  in  both  cases  we  see 
a  characteristic  feature  of  detail,  namely  that  the  "spread"  of  the 
curve  is  greater  in  the  second  wave  than  in  the  first,  that  is  to 


20  25  30 

Length  of  fish,  in  cm. 

Fig.  24.     Variability  of  length  of  body  in  a  sample  of  Plaice. 

say  (in  the  case  of  the  fish)  in  the  older  as  well  as  larger  series. 
Over  the  reason  for  this  phenomenon,  which  is  simple  and  all  but 
obvious,  we  need  not  pause. 

It  is  evident,  then,  that  in  this  case  of  "dimorphism,"  the  tails 
of  the  one  group  of  earwigs  (which  Bateson  calls  the  "high  males") 
have  either  grown  faster,  or  have  been  growing  for  a  longer  period 
of  time,  than  those  of  the  "low  males."  If  we  could  be  certain 
that  the  whole  random  sample  of  earwigs  were  of  one  and  the 
same  age,  then  we  should  have  to  refer  the  phenomenon  of  di- 
morphism to  a  physiological  phenomenon,  simple  in  kind  (however 
remarkable  and  unexpected) ;   viz.  that  there  were  two  alternative 


106  THE   RATE   OF   GROWTH  [ch. 

values,  very  different  from  one  another,  for  the  mean  velocity  of 
growth,  and  that  the  individual  earwigs  varied  around  one  or 
other  of  these  mean  values,  in  each  case  according  to  the  law  of 
probabilities.  But  on  the  other  hand,  it  we  could  beheve  that 
the  two  groups  of  earwigs  were  of  different  ages,  then  the  pheno- 
menon would  be  simphcity  itself,  and  there  would  be  no  more  to 
be  said  about  it*. 

Before  we  pass  from  the  subject  of  the  relative  rate  of  growth 
of  different  parts  or  organs,  we  may  take  brief  note  of  the  fact 
that  various  experiments  have  been  made  to  determine  whether 
the  normal  ratios  are  maintained  under  altered  circumstances  of 
nutrition,  and  especially  in  the  case  of  partial  starvation.  For 
instance,  it  has  been  found  possible  to  keep  young  rats  alive  for 
many  weeks  on  a  diet  such  as  is  just  sufficient  to  maintain  life 
without  permitting  any  increase  of  weight.  The  rat  of  three 
weeks  old  weighs  about  25  gms.,  and  under  a  normal  diet  should 
weigh  at  ten  weeks  old  about  150  gms.,  in  the  male,  or  115  gms. 
in  the  female ;  but  the  underfed  rat  is  still  kept  at  ten  weeks  old 
to  the  weight  of  25  gms.  Under  normal  diet  the  proportions  of 
the  body  change  very  considerably  between  the  ages  of  three  and 
ten  weeks.  For  instance  the  tail  gets  relatively  longer ;  and  even 
when  the  total  growth  of  the  rat  is  prevented  by  underfeeding, 
the  form  continues  to  alter  so  that  this  increasing  length  of  the 
tail  is  still  manifest  |. 

*  I  do  not  say  that  the  assumption  that  these  two  groups  of  earwigs  were  of 
different  ages  is  altogether  an  easy  one;  for  of  course,  even  in  an  insect  whose 
metamorphosis  is  so  simple  as  the  earwig's,  consisting  only  in  the  acquisition  of 
wings  or  wing-cases,  we  usually  take  it  for  granted  that  growth  proceed?  no  more 
after  the  final  stage,  or  "adult  form"  is  attained,  and  further  that  this  adult  form 
is  attained  at  an  approximate^  constant  age,  and  constant  magnitude.  But  even 
if  we  are  not  permitted  to  think  that  the  earwig  may  have  grown,  or  moulted, 
after  once  the  elytra  were  produced,  it  seems  to  me  far  from  impossible,  and  far 
from  unlikely,  that  prior  to  the  appearance  of  the  elytra  one  more  stage  of  growth, 
or  one  more  moult  took  place  in  some  cases  than  in  others:  for  the  number  of 
moults  is  known  to  be  variable  in  many  species  of  Orthoptera.  Unfortunately 
Bateson  tells  us  nothing  about  the  sizes  or  total  lengths  of  his  earwigs;  but  his 
figures  suggest  that  it  was  bigger  earwigs  that  had  the  longer  tails ;  and  that  the 
rate  of  growth  of  the  tails  had  had  a  certain  definite  ratio  to  that  of  the  bodies, 
but  not  necessarily  a  simple  ratio  of  equality. 

■j-  Jackson,  C.  M.,  J.  of  Exp.  Zool.  xix,  1915,  p.  99;  cf.  also  Hans  Aron,  Unters. 


Ill]  THE  EFFECT  OF  TEMPERATURE  107 


Full-fed  Rats. 

Age  in 

Length  of      Length  of 

Total 

weeks 

body  (mm.)      tail  (m.) 

length 

Oq  of  tail 

0 

48-7                16-9 

65-6 

25-8 

1 

64-5                29-4 

93-9 

31-3 

3 

90-4               59-1 

149-5 

39-5 

6 

128-0             110-0 

238-0 

46-2 

10 

1730             150-0 

Underfed  Rats 

323-0 

46-4 

6 

98-0              72-3 

170-3 

42-5 

10 

99-6              83-9 

183-5 

45-7 

Again  as  physiologists  have  long  been  aware,  there  is  a  marked 
difference  in  the  variation  of  weight  of  the  different  organs, 
according  to  whether  the  animal's  total  weight  remain  constant, 
or  be  caused  to  diminish  by  actual  starvation  ;  and  further  striking 
differences  appear  when  the  diet  is  not  only  scanty,  but  ill-balanced. 
But  these  phenomena  of  abnormal  growth,  however  interestmg 
from  the  physiological  view,  are  of  little  practical  importance  to 
the  morphologist. 

The  effect  of  temperature*. 

The  rates  of  growth  which  we  have  hitherto  dealt  with  arc 
based  on  special  investigations,  conducted  under  particular  local 
conditions.  For  instance,  Quetelet's  data,  so  far  as  we  have  used 
them  to  illustrate  the  rate  of  growth  in  man,  are  drawn  from  his 
study  of  the  population  of  Belgium.  But  apart  from  that 
"fortuitous"  individual  variation  which  we  have  already  con- 
sidered, it  is  obvious  that  the  normal  rate  of  growth  will  be  found 
to  vary,  in  man  and  in  other  animals,  just  as  the  average  stature 
varies,  in  different  localities,  and  in  different  "races."  This 
phenomenon  is  a  very  complex  one,  and  is  doubtless  a  resultant 
of  many  undefined  contributory  causes ;  but  we  at  least  gain 
something  in  regard  to  it,  when  we  discover  that  the  rate  of  growth 
is  directly  affected  by  temperature,  and  probably  by  other  physical 

iiber  die  Beeinfliissung  der  Wachstum  durch  die  Ernahrung,  Berl.  klin.  Wochenhl. 
LI,  pp.  972-977,  1913,  etc. 

*  The  temperature  limitations  of  life,  and  to  some  extent  of  growth,  are  summar- 
ised for  a  large  number  of  species  by  Davenport,  ^x^er.  Mor2')}iology ,  cc.  viii,  xviii, 
and  by  Hans  Przibram,  Exp.  Zoologie,  iv,  c.  v. 


108  THE   KATE   OF   GROWTH  [ch. 

conditions.  Reaumur  was  the  first  to  shew,  and  the  observation 
was  repeated  by  Bonnet*,  that  the  rate  of  growth  or  development 
of  the  chick  was  dependent  on  tem.perature,  being  retarded  at 
temperatures  below  and  somewhat  accelerated  at  temperatures 
above  the  normal  temperature  of  incubation,  that  is  to  say  the 
temperature  of  the  sitting  hen.  In  the  case  of  plants  the  fact 
that  growth  is  greatly  affected  by  temperature  is  a  matter  of 
familiar  knowledge;  the  subject  was  first  carefully  studied  by 
Alphonse  De  Candolle,  and  his  results  and  those  of  his  followers 
are  discussed  in  the  textbooks  of  Botany f. 

That  variation  of  temperature  constitutes  only  one  factor  in  determining 
the  rate  of  growth  is  admirably  illustrated  in  the  case  of  the  Bamboo.  It  has 
been  stated  (by  Lock)  that  in  Ceylon  the  rate  of  growth  of  the  Bamboo  is 
directly  proportional  to  the  humidity  of  the  atmosphere:  and  again  (by 
Shibata)  that  in  Japan  it  is  directly  proportional  to  the  temperature.  The 
two  statements  have  been  ingeniously  and  satisfactorily  reconciled  by 
BlackmanJ,  who  suggests  that  in  Ceylon  the  temperature-conditions  are 
all  that  can  be  desired,  but  moisture  is  apt  to  be  deficient:  while  in  Japan 
there  is  rain  in  abundance  but  the  average  temperature  is  somewhat  too  low. 
So  that  in  the  one  country  it  is  the  one  factor,  and  in  the  other  country  it  is 
the  other,  which  is  essentially  variable. 

The  annexed  diagram  (Fig.  25),  shewing  the  growth  in  length 
of  the  roots  of  some  common  plants  during  an  identical  period 
of  forty-eight  hours,  at  temperatures  varying  from  about  14°  to 
37°  C,  is  a  sufiicient  illustration  of  the  phenomenon.  We  see  that 
in  all  cases  there  is  a  certain  optimum  temperature  at  which  the 
rate  of  growth  is  a  maximum,  and  we  can  also  see  that  on  either 
side  of  this  optimum  temperature  the  acceleration  of  growth, 
positive  or  negative,  with  increase  of  temperature  is  rapid,  while 
at  a  distance  from  the  optimum  it  is  very  slow.  From  the 
data  given  by  Sachs  and  others,  we  see  further  that  this  optimum 
temperature  is  very  much  the  same  for  all  the  common  plants  of 
our  own  climate  which  have  as  yet  been  studied ;    in  them  it  is 

*  Reaumur :  Uart  de  faire  colore  et  elever  en  toufe  saison  des  oiseaux  domestiques, 
foil  ixir  le  moyeii  de  la  chaleur  du  fumier,  Paris,  1749. 

f  Cf.  (int.  al.)  de  Vries,  H.,  Materiaux  pour  la  connaissance  de  rinfluence  de 
la  temperature  sur  les  plantes,  Arch.  Neerl.  v,  385-401,  1870.  Koppen,  Warme 
und  Pflanzenwachstum,  Bull.  Soc.  Imp.  Nat.  Moscou.  XLiii,  pp.  41-110,  1870. 

X  Blackman,  F.  F.,  Ann.  of  Botany,  xix,  p.  281,  1905. 


Ill 


THE  EFFECT  OF  TEMPEKATURE 


109 


somewhere  about  26°  C.  (or  say  77°  F.),  or  about  the  temperature 
of  a  warm  summer's  day ;  while  it  is  found,  very  naturally,  to  be 
considerably  higher  in  the  case  of  plants  such  as  the  melon  or  the 
maize,  which  are  at  home  in  warmer  regions  that  our  own. 

In  a  large  number  of  physical  phenomena,  and  in  a  very  marked 
degree  in  all  chemical  reactions,  it  is  found  that  rate  of  action  is 
affected,  and  for  the  most  part  accelerated,  by  rise  of  temperature  ; 


14°16    18    20    22    24    26    28'   30    32    34    36    38  40° 

Temp. 

Fig.  25.     Relation  of  rate  of  growth  to  temperature  in  certain  plants. 
(From  Sachs's  data.) 

and  this  effect  of  temperature  tends  to  follow  a  definite  "ex- 
ponential" law,  which  holds  good  within  a  considerable  range  of 
temperature,  but  is  altered  or  departed  from  when  we  pass  beyond 
certain  normal  hmits.  The  law,  as  laid  down  by  van't  Hoff  for 
chemical  reactions,  is,  that  for  an  interval  of  n  degrees  the  velocity 
varies  as  x'^,  x  being  called  the  "temperature  coefficient"*  for  the 
reaction  in  question. 

*  For  various  instances  of  a  "temperature  coefficient"  in  physiological  pro- 
cesses, see  Kanitz,  Zeitschr.  f.  Elektrochemie,  1907,  p.  707;  Biol.  Centralhl.  xxvii, 
p.  11,  1907;  Hertzog,  R.  0.,  Temperatureinfluss  auf  die  Entwicklungsgesch- 
windigkeit  der  Organismen,  Zeitschr.  f.  Elektrochemie,  xi,  p   820,  1905;    Krogh, 


110  THE  RATE   OF   GROWTH  [ch. 

Van't  Hoffs  law,  which  has  become  a  fundamental  principle 
of  chemical  mechanics,  is  likewise  applicable  (with  certain  qualifica- 
tions) to  the  phenomena  of  vital  chemistry ;  and  it  follows  that, 
on  very  much  the  same  lines,  we  may  speak  of  the  "temperature 
coefficient"  of  growth.  At  the  same  time  we  must  remember 
that  there  is  a  very  important  difference  (though  we  can  scarcely 
call  it  a  fundamental  one)  between  the  purely  physical  and  the 
physiological  phenomenon,  in  that  in  the  former  we  study  (or 
seek  and  profess  to  study)  one  thing  at  a  time,  while  in  the  latter 
we  have  always  to  do  with  various  factors  which  intersect  and 
interfere ;  increase  in  the  one  case  (or  change  of  any  kind)  tends 
to  be  continuous,  in  the  other  case  it  tends  to  be  brought  to  arrest. 
This  is  the  simple  meaning  of  that  Law  of  Optimmn,  laid  down  by 
Errera  and  by  Sachs  as  a  general  principle  of  physiology :  namely 
that  every  physiological  process  which  varies  (like  growth  itself) 
with  the  amount  or  intensity  of  some  external  influence,  does  so 
according  to  a  law  in  which  progressive  increase  is  followed  by 
progressive  decrease ;  in  other  words  the  function  has  its  optimum 
condition,  and  its  curve  shews  a  definite  maximum^.  In  the  case 
of  temperature,  as  Jost  puts  it,  it  has  on  the  one  hand  its  accelerat- 
ing effect  which  tends  to  follow  van't  Hoff's  law.  But  it  has  also 
another  and  a  cumulative  eft'ect  upon  the  organism :  "  Sie  schadigt 
oder  sie  ermiidet  ihn,  und  je  hoher  sie  steigt,  desto  rascher  macht 
sie  die  Schadigung  geltend  und  desto  schneller  schreitet  sie  voran." 
It  would  seem  to  be  this  double  effect  of  temperature  in  the  case 
of  the  organism  which  gives  us  our  "optimum"  curves,  which  are 
the  expression,  accordingly,  not  of  a  primary  phenomenon,  but 
of  a  more  or  less  complex  resultant.  Moreover,  as  Blackman  and 
others  have  pointed  out,  our  '"optimum"  temperature  is  very 
ill-defined  until  we  take  account  also  of  the  duration  of  our  experi- 
ment ;  for  obviously,  a  high  temperature  may  lead  to  a  short, 
but  exhausting,  spell  of  rapid  growth,  while  the  slower  rate 
manifested  at  a  lower  temperature  may  be  the  best  in  the  end. 

Quantitative  Relation  between  Temperature  and  Standard  Metabolism,  l7it. 
Zeifschr.  J.  physik.-chem.  Biologie,  i,  p.  491,  1914;  Piitter,  A.,  Ueber  Temperatur- 
koefficienten,  Zeiischr.  f.  allgern.  Phi/siol.  xvi,  p.  574,  1914.  Also  Cohen. 
Physical  Chemistry  for  Physicians  and  Biologists  (English  edition),  1903;  Pike, 
F.  H.,  and  Scott,  E.  L.,  The  Regulation  of  the  Physico-chemical  Condition  of  the 
Orsranism,  American  Naturalist,  Jan.  1915,  and  various  papers  quoted  therein. 


Ill]  THE  EFFECT  OF  TEMPERATURE  111 

The  mile  and  the  hundred  yards  are  won  by  different  runners ; 
and  maximum  rate  of  working,  and  maximum  amount  of  work 
done,  are  two  very  different  things*. 

In  the  case  of  maize,  a  certain  series  of  experiments  shewed  that 
the  growth  in  length  of  the  roots  varied  with  the  temperature  as 
followsl : 


Temperature 
°C. 

Growth  in  48  hours 
mm. 

180 

11 

23-5 

10-8 

26-6 

29-6 

28-5 

26-5 

30-2 

64-6 

33-5 

69-5 

36-5 

20-7 

Let  us  write  our  formula 

in 

the 

form 

V 

(t+ 

n)  _ 

■X''  . 

Yt 

Then  choosing  two  values  out  of  the  above  experimental  series 
(say  the  second  and  the  second-last),  we  have  /  =  23-5,  n  =  10, 
and  F,   F'  =  10-8  and  69-5  respectively. 

69'5 
Accordingly  YrTo  =  6-4  =  x^^. 

Therefore  ^f^ '  ^^  '^^^^  =  ^°g  ^• 

And,  X  =  1-204  (for  an  interval  of  1°  C). 

This  first  approximation  might  be  considerably  improved  by 
taking  account  of  all  the  experimental  values,  two  only  of  which 
we  have  as  yet  made  use  of ;  but  even  as  it  is,  we  see  by  Fig.  26 
that  it  is  in  very  fair  accordance  with  the  actual  results  of 
observation,  within  those  particular  limits  of  temperature  to  which 
the  experiment  is  confined. 

*  Cf.  Errera,  L.,  UOptimmn,  1896  {Rec.  d'Oeuvres,  Physiol,  generate,  pp.  338-368, 
1910);  Sachs,  Physiologie  d.  Pflanzen,  1882,  p.  233;  Pfeffer,  Pftunzenphysiologie, 
ii,  p.  78,  1904;  and  cf.  Jost,  Ueber  die  Reactionsgeschwindigkeit  im  Organismus, 
Biol.  Centralbl.  xxvi,  pp.  225-244,   1906. 

t  After  Koppen,  Bull.  Hoc.  Nat.  Moscou,  xliii,  pp.  41-110,  1871. 


112 


THE   RATE   OF   GROWTH 


[CH. 


For  an  experiment  on  Lupinus  albus,  quoted  by  Asa  Gray*, 
I  have  worked  out  the  corresponding  coefficient,  but  a  little  more 
carefully.  Its  value  I  find  to  be  1-16,  or  very  nearly  identical 
with  that  we  have  just  found  for  the  maize ;  and  the  correspondence 
between  the  calculated  curve  and  the  actual  observations  is  now 
a  close  one. 


18       20        22 


24        26         28 
Temperature 


30 


Fig.  26. 


Relation  of  rate  of  growth  to  temijerature  in  Maize, 
values  (after  Koppen).  and  calculated  curve. 


34°C. 


Observed 


Since  the  above  paragraphs  were  wi'itten,  new  data  have  come  to  hand. 
Miss  I.  Leitch  has  made  careful  observations  of  the  rate  of  growth  of  rootlets 
of  the  Pea ;  and  I  have  attempted  a  further  analysis  of  her  principal  resultsy. 
In  Fig.  27  are  shewn  the  mean  rates  of  growth  (based  on  about  a  hundred 
experiments)  at  some  thirty-four  different  temperatures  between  0-8°  and 
29-3°,  each  experiment  lasting  rather  less  than  twenty-four  hours.  Working 
out  the  mean  temperature  coefficient  for  a  great  many  combinations  of  these 
values,  I  obtain  a  value  of  1-092  per  C.°,  or  2-41  for  an  interval  of  10°,  and 
a  mean  value  for  the  whole  series  showing  a  rate  of  growth  of  just  about 
1  mm.  per  hotir  at  a  temperature  of  20°.  My  curve  in  Fig.  27  is  drawn  from 
these  determinations ;  and  it  will  be  seen  that,  while  it  is  by  no  means  exact 
at  the  lower  temperatures,  and  will  of  course  fail  us  altogether  at  very  high 

*  Botany,  p.  387. 

t  Leitch,  I.,  Some  Experiments  on  the  Influence  of  Temperature  on  the  Rate 
of  Growth  in  Pisum  sativum,  Ann.  of  Botany,  xxx,  pp.  25-46,  1916.  (Cf.  especiallj' 
Table  III,  p.  45.) 


IIIj 


THE  EFFECT  OF  TEMPERATURE 


113 


temperatures,  yet  it  serves  as  a  very  satisfactory  guide  to  the  relations  between 
rate  and  temperature  within  the  ordinary  limits  of  healthy  growth.  Miss 
Leitch  holds  that  the  curve  is  not  a  van't  Hoff  curve ;  and  this,  in  strict  accuracy, 
we  need  not  dispute.  But  the  phenomenon  seems  to  me  to  be  one  into  which 
the  van't  Hoff  ratio  enters  largely,  though  doubtless  combined  with  other 
factors  which  we  cannot  at  present  determine  or  eliminate. 


4° 


12°       16°       20° 
Temperature 


24° 


zu 

/• 

— 

18 

/ 

16 

/ 

14 

/ 

_ 

/ 

12 

V 

- 

/ 

10 

/ 

— 

/ 

8 

V. 

/ 

/ 

6 

• 
/ 

/ 

/ 

/ 

— 

4 

^ 

^* 

• 

^ 

— 

2 

• 

I 

y 

-^ 

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'per 
hour 

2-0 


1-8 


1-6 


28°      32' 


4  2 


2   p 


1-0   ^ 

•8  S 

O 

•6 
•4 
•2, 
0 


Fig.  27.     Relation  of  rate  of  growth  to  temperature  in  rootlets 
Pea.     (From  Miss  I.  Leitch's  data.) 


of 


While  the  above  results  conform  fairly  well  to  the  law  of  the 
temperature  coefficient,  it  is  evident  that  the  imbibition  of  water 
plays  so  large  a  part  in  the  process  of  elongation  of  the  root  or 
stem  that  the  phenomenon  is  rather  a  physical  than  a  chemical 
one;  and  on  this  account,  as  Blackman  has  remarked,  the  data 
commonly  given  for  the  rate  of  growth  in  plants  are  apt  to  be 


114  THE  EATE   OF   GROWTH  [ch. 

irregular,  and  sometimes  (we  might  even  say)  misleading  *.  The 
fact  also,  which  we  have  already  learned,  that  the  elongation  of  a 
shoot  tends  to  proceed  by  jerks,  rather  than  smoothly,  is  another 
indication  that  the  phenomenon  is  not  purely  and  simply  a 
chemical  one.  We  have  abundant  illustrations,  however,  among 
animals,  in  which  we  may  study  the  temperature  coefficient  under 
circumstances  where,  though  the  phenomenon  is  always  compli- 
cated by  osmotic  factors,  true  metabolic  growth  or  chemical 
combination  plays  a  larger  role.  Thus  Mile.  Maltaux  and  Professor 
Massartf  have  studied  the  rate  of  division  in  a  certain  flagellate, 
Cliilomonas  faramoecium,  and  found  the  process  to  take  29  minutes 
at  15°  C,  12  at  25°,  and  only  5  minutes  at  35°  C.  These  velocities 
are  in  the  ratio  of  1  :  2-4  :  5-76,  which  ratio  corresponds  precisely 
to  a  temperature  coefficient  of  2-4  for  each  rise  of  10°,  or  about 
1*092  for  each  degree  centigrade. 

By  means  of  this  principle  we  may  throw  light  on  the  apparently 
comphcated  results  of  many  experiments.  For  instance.  Fig.  28 
is  an  illustration,  which  has  been  often  copied,  of  0.  Hertwig's 
work  on  the  effect  of  temperature  on  the  rate  of  development  of 
the  tadpolef , 

From  inspection  of  this  diagram,  we  see  that  the  time  taken 
to  attain  certain  stages  of  development  (denoted  by  the  numbers 
III-VII)  was  as  follows,  at  20°  and  at  10°  C,  respectively. 

At  20°  At  10° 


Stage  III 

2-0 

6-5  days 

„     IV 

2-7 

8-1     „ 

„     V 

3-0 

10-7     „ 

„     VI 

40 

13-5     „ 

„     VII 

50 

16-8     „ 

Total 

16-7 

55-6     „ 

That  is  to  say,  the  time  taken  to  produce  a  given  result  at 

*  Blackman,  F.  F.,  Presidential  Address  in  Botany,  Brit.  Ass.  Dublin,  1908. 

f  Rec.  de  Vljist.  Bot.  de  Bruxelles,  vi,  1906. 

X  Hertwig,  0.,  Einfluss  der  Temperatur  auf  die  Entwicklung  von  Bana  fusca 
und  R.  esculenia,  Arch.  f.  mikrosk.  Anat.  li,  p.  319,  1898.  Cf.  also  Bialaszewicz, 
K.,  Beitrage  z.  Kenntniss  d.  Wachsthumsvorgange  bei  Amphibienembryonen, 
Bull.  Acad.  Sci.  de  Cracovie,  p.  783,  1908 ;  Abstr.  in  Arch.  f.  Entwicklung smech. 
xxvm,  p.  160,  1909. 


Ill] 


THE   EFFECT   OF   TEMPERATURE 


115 


10°  was  (on  the  average)  somewhere  about  55-6/16-7,  or  3-33, 
times  as  long  as  was  required  at  20°. 


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■ — ■ 

TemrLpXa,lu%e     CentigriacLe 

2#-°  ZJ"  ii"  H'  iO"  19'  18°  It'   16'   l{>'    IV  13''    n"   II'    10°    9°    a"     '^    6° 

Fig.  28.  Diagram  shewing  time  taken  (in  days),  at  various  temperatures  (°C.), 
to  reach  certain  stages  of  development  in  the  Frog :  viz.  I,  gastrula ;  II, 
medullary  plate;  III,  closure  of  medullary  folds;  IV,  tail-bud;  V,  tail  and 
gills ;  VI,  tail-fin ;  VII,  operculum  beginning ;  VIII,  do.  closing ;  IX,  first 
appearance  of  hind-legs.     (From  Jenkinson,  after  0.  Hertwig,  1898.) 

We  may  then  put  our  equation  again  in  the  simple  form, 
xi"  -  333. 


8—2 


116 


THE   RATE   OF   GROWTH 


[CH. 


Or, 

Therefore 


and 


10  log  X  =  log  3-33  -  -52244. 
log  X  -  -05224, 
x=  1-128. 


That  is  to  say,  between  the  intervals  of  10°  and  20°  C,  if  it 
take  m  days,  at  a  certain  given  temperature,  for  a  certain  stage 
of  development  to  be  attained,  it  will  take  m  x  1-128"  days, 
when  the  temperature  is  n  degrees  less,  for  the  same  stage  to 
be  arrived  at. 


25'C. 


20 


0° 


15°  10° 

Temperature 

Fig.  29.     Calculated  values,  corresponding  to  preceding  figure. 


Fig.  29  is  calculated  throughout  from  this  value;  and  it  will 
be  seen  that  it  is  extremely  concordant  with  the  original  diagram, 
as  regards  all  the  stages  of  development  and  the  whole  range  of 
temperatures  shewn :  in  spite  of  the  fact  that  the  coefficient  on 
which  it  is  based  was  derived  by  an  easy  method  from  a  very  few 
points  in  the  original  curves. 


Ill]  THE   EFFECT   OF   TEMPERATURE  117 

Karl  Peter*,  experimenting  chiefly  on  echinoderm  eggs,  and 
also  making  use  of  Hertwig's  experiments  on  young  tadpoles, 
gives  the  normal  temperature  coefficients  for  intervals  of  10°  C. 
(commonly  written  Q^q)  as  follows. 

Sphaerechinus        ...         ...         ...         2-15, 

Echinus       2-13, 

Rana  2-86. 

These  values  are  not  only  concordant,  but  are  evidently  of  the 
same  order  of  magnitude  as  the  temperature-coefficient  in  ordinary 
chemical  reactions.  Peter  has  also  discovered  the  very  interesting 
fact  that  the  temperature-coefficient  alters  with  age,  usually  but 
not  always  becoming  smaller  as  age  increases. 

Sphaerechinus ;    Segmentation  Q^"  =  2-29, 

Later  stages  ,,     2-03. 

Echinus;              Segmentation  ,,     2*30, 

Later  stages  ,,      2-08. 

Rana;                   Segmentation  ,,     2-23, 

Later  stages  ,,     3-34. 

Furthermore,  the  temperature  coefficient  varies  with  the 
temperature,  diminishing  as  the  temperature  rises, — a  rule  which 
van't  Hoff  has  shewn  to  hold  in  ordinary  chemical  operations. 
Thus,  in  Rana  the  temperature  coefficient  at  low  temperatures 
may  be  as  high  as  5-6 :  which  is  just  another  way  of  saying  that 
at  low  temperatures  development  is  exceptionally  retarded. 

In  certain  fish,  such  as  plaice  and  haddock,  I  and  others  have 
found  clear  evidence  that  the  ascending  curve  of  growth  is  subject 
to  seasonal  interruptions,  the  rate  during  the  winter  months 
being  always  slower  than  in  the  months  of  summer :  it  is  as  though 
we  superimposed  a  periodic,  annual,  sine-curve  upon  the  continuous 
curve  of  growth.  And  further,  as  growth  itself  grows  less  and  less 
from  year  to  year,  so  will  the  difference  between  the  winter  and 
the  summer  rate  also  grow  less  and  less.     The  fluctuation  in  rate 

*  Der  Grad  der  Beschleunigung  tierischer  Entwickelung  durch  erhcihte 
Temperatur,  A.  f.  Entiv.  Mech.  xx,  p.  130,  1905.  More  recently,  Bialaszewicz 
has  determined  the  coefficient  for  the  rate  of  segmentation  in  Rana  as  being 
2-4  per  10°  C. 


118  THE   RATE   OF   GROWTH  [ch. 

will  represent  a  vibration  which  is  gradually  dying  out ;  the  ampli- 
tude of  the  sine-curve  will  gradually  diminish  till  it  disappears ; 
in  short,  our  phenomenon  is  simply  expressed  by  what  is  known 
as  a  "damped  sine-curve."  Exactly  the  same  thing  occurs  in 
man,  though  neither  in  his  case  nor  in  that  of  the  fish  have  we 
sufficient  data  for  its  complete  illustration. 

We  can  demonstrate  the  fact,  however,  in  the  case  of  man  by 
the  help  of  certain  very  interesting  measurements  which  have 
been  recorded  by  Daffner  *,  of  the  height  of  German  cadets, 
measured  at  half-yearly  intervals. 

Growth  in  height  of  German  military  Cadets,  in  half-yearly 
periods.     {Dajfner.) 

Increment  in  cm. 


Number 
observed 

iXC 

IgllU    lit    V.C 

A 

HI/. 

Winter 

^-year 

Summer 
•1-year 

■* 

Age 

October 

April 

October 

Year 

12 

11-12 

139-4 

141-0 

143-3 

"l-6 

2-3 

3-9 

80 

12-13 

1430 

144-5 

147-4 

1-5 

2-9 

4-4 

146 

13-14 

147-5 

149-5 

152-5 

2-0 

3-0 

5-0 

162 

14-15 

152-2 

155-0 

158-5 

2-5 

3-5 

6-0 

162 

15-16 

158-5 

160-8 

163-8 

2-3 

3-0 

5-3 

150 

16-17 

163-5 

165-4 

167-7 

1-9 

2-3 

4-2 

82 

17-18 

167-7 

168-9 

170-4 

1-2 

1-5 

2-7 

22 

18-19 

169-8 

170-6 

171-5 

0-8 

0-9 

1-7 

6 

19-20 

170-7 

171-1 

171-5 

0-4 

0-4 

0-8 

In  the  accompanying  diagram  (Fig.  30)  the  half-yearly  incre- 
ments are  set  forth,  from  the  above  table,  and  it  will  be  seen  that 
they  form  two  even  and  entirely  separate  series.  The  curve 
joining  up  each  series  of  points  is  an  acceleration-curve;  and  the 
comparison  of  the  two  curves  gives  a  clear  view  of  the  relative 
rates  of  growth  during  winter  and  summer,  and  the  fluctuation 
which  these  velocities  undergo  during  the  years  in  question.  The 
dotted  Hne  represents,  approximately,  the  acceleration-curve  in 
its  continuous  fluctuation  of  alternate  seasonal  decrease  and 
increase. 

In  the  case  of  trees,  the  seasonal  fluctuations  of  growthl  admit 

*  Das  Wachstum  des  Menschen,  p.  329,  1902. 

f  The  diurnal  periodicity  is  beautifully  shewn  in  the  case  of  the  Hop  by  Joh. 
Schmidt  (C  E.  du  Laboratoire  de  Garlsberg,  x,  pp.  235-248,  Copenhague,   1913). 


Ill] 


THE   EFFECT   OF   TEMPERATURE 


119 


of  easy  determination,  and  it  is  a  point  of  considerable  interest 
to  compare  the  phenomenon  in  evergreen  and  in  deciduous  trees. 
I  happen  to  have  no  measurements  at  hand  with  which  to  make 
this  comparison  in  the  case  of  our  native  trees,  but  from  a  paper 
by  Mr  Charles  E.  Hall*  I  have  compiled  certain  mean  values  for 
growth  in  the  climate  of  Uruguay. 


19       20 

years 

Fig.  30.     Half-yearly  increments  of  growth,  in  cadets  of  various  ages. 
(From  Daffner's  data.) 

Mean  monthly  increase  in  Girth  of  Evergreen  a?id  Deciduous  Trees, 
at  San  Jorge,  Uruguay.  {After  C.  E.  Hall.)  Values  expressed 
as  'percentages  of  total  annual  increase. 

Jan.  Feb.  Mar.  Apr.  May  June  July  Aug.  Sept.  Oct.  Nov.  Dec. 

Evergreens    9-1  8-8  8-6     8-9     7-7  5-4  4-3  6-0     9-1  11 -1  10-8  10-2 
Deciduous 

trees       ...  20-3  14-6  90     2-3     0-8  0-3  0-7  1-3     3-5  9-9  16-7  21-0 

The  measurements  taken  were  those  of  the  girth  of  the  tree, 
in  mm.,  at  three  feet  from  the  ground.  The  evergreens  included 
species  of  Pinus,  Eucalyptus  and  iVcacia ;  the  deciduous  trees 
included  Quercus,  Populus,  Robinia  and  Meha.  I  have  merely 
taken  mean  values  for  these  two  groups,  and  expressed  the 
monthly  values  as  percentages  of  the  mean  annual  increase.  The 
result  (as  shewn  by  Fig.  31)  is  very  much  what  we  might  have 
expected.  Th^  growth  of  the  deciduous  trees  is  completely 
arrested  in  winter-time,  and  the  arrest  is  all  but  complete  over 

*  Trans.  Botan.  Soc.  Edinburgh,  xvm,  1891,  p.  456. 


120 


THE   RATE   OF   GROWTH 


[CH. 


a  considerable  period  of  time ;  moreover,  during  the  warm  season, 
the  monthly  values  are  regularly  graded  (approximately  in  a 
sine-curve)  with  a  clear  maximum  (in  the  southern  hemisphere) 
about  the  month  of  December.  In  the  evergreen  trees,  on  the 
other  hand,  the  amplitude  of  the  periodic  wave  is  very  much 
less ;  there  is  a  notable  amount  of  growth  all  the  year  round, 
and,  while  there  is  a  marked  diminution  in  rate  during  the  coldest 
months,  there  is  a  tendency  towards  equahty  over  a  considerable 


Fig.  31.     Periodic  annual  fluctuation  in  rate  of  growth  of  trees  (in  the 
southern  hemisphere). 

part  of  the  warmer  season.  It  is  probable  that  some  of  the 
species  examined,  and  especially  the  pines,  were  definitely  retarded 
in  growth,  either  by  a  temperature  above  their  optimum,  or  by 
deficiency  of  moisture,  during  the  hottest  period  of  the  year; 
with  the  result  that  the  seasonal  curve  in  our  diagram  has  (as  it 
were)  its  region  of  maximum  cut  off. 

In  the  case  of  trees,  the  seasonal  periodicity  of  growth  is  so 
well  marked  that  we  are  entitled  to  make  use  of  the  phenomenon 
in  a  converse  way,  and  to  draw  deductions  as  to  variations  in 


Ill]  THE   EFFECT   OF   CLIMATE  121 

climate  during  past  years  from  the  record  of  varying  rates  of 
growth  which  the  tree,  by  the  thickness  of  its  annual  rings,  has 
preserved  for  us.  Mr  A.  E.  Douglass,  of  the  University  of 
Arizona,  has  made  a  careful  study  of  this  question*,  and  I  have 
received  (through  Professor  H.  H.  Turner  of  Oxford)  some  measure- 
ments of  the  average  width  of  the  successive  annual  rings  in  "  yellow 
pine,"  500  years  old,  from  Arizona,  in  which  trees  the  annual 
rings  are  very  clearly  distinguished.  From  the  year  1391  to  1518, 
the  mean  of  two  trees  was  used;  from  1519  to  1912,  the  mean  of 
five;  and  the  means  of  these,  and  sometimes  of  larger  numbers, 
were  found  to  be  very  concordant.  A  correction  was  applied  by 
drawing  a  long,  nearly  straight  line  through  the  curve  for  the 
whole  period,  which  line  was  assumed  to  represent  the  slowly 
diminishing  mean  width  of  ring  accompanying  the  increase  of 
size,  or  age,  of  the  tree ;  and  the  actual  growth  as  measured  was 
equated  with  this  diminishing  mean.  The  figures  used  give, 
accordingly,  the  ratio  of  the  actual  growth  in  each  year  to  the 
mean  growth  corresponding  to  the  age  or  magnitude  of  the  tree 
at  that  epoch. 

It  was  at  once  manifest  that  the  rate  of  growth  so  determined 
shewed  a  tendency  to  fluctuate  in  a  long  period  of  between  100  and 
200  years.  I  then  smoothed  in  groups  of  100  (according  to  Gauss's 
method)  the  yearly  values,  so  that  each  number  thus  found 
represented  the  mean  annual  increase  during  a  century:  that  is 
to  say,  the  value  ascribed  to  the  year  1500  represented  the  average 
annual  growth  during  the  whole  period  between  1450  and  1550, 
and  so  on.  These  values  give  us  a  curve  of  beautiful  and  surprising 
smoothness,  from  which  we  seem  compelled  to  draw  the  direct 
conclusion  that  the  climate  of  Arizona,  during  the  last  500  years, 
has  fluctuated  .with  a  regular  periodicity  of  almost  precisely  150 
years.     Here  again  we  should  be  left  in  doubt  (so  far  as  these 

*  I  had  not  received,  when  this  was  written,  Mr  Douglass's  paper,  On  a  method 
of  estimating  Rainfall  by  the  Growth  of  Trees,  Bull.  Arner.  Geograph.  Soc.  xlvi. 
pp.  321-335,  1914.  Mr  Douglass  does  not  fail  to  notice  the  long  period  here 
described;  but  he  lays  more  stress  on  the  occurrence  of  shorter  cycles  (of  11,  21 
and  33  years),  well  known  to  meteorologists.  Mr  Douglass  is  inchned  (and  I  think 
rightly)  to  correlate  the  variations  in  growth  directly  with  fluctuations  in  rainfall, 
that  is  to  say  with  alternate  periods  of  moisture  and  aridity;  but  he  points  out 
that  the  temperature  curves  (and  also  the  sunspot  curves)  are  markedly  similar. 


*i    I— t 


O     o 


b  ;z; 


CH.  Ill]  THE   EFFECT   OF   CLIMATE  123 

observations  go)  whether  the  essential  factor  be  a  fluctuation  of 
temperature  or  an  alternation  of  moisture  and  aridity ;  but  the 
character  of  the  Arizona  climate,  and  the  known  facts  of  recent 
years,  encourage  the  belief  that  the  latter  is  the  more  direct  and 
more  important  factor. 

It  has  been  often  remarked  that  our  common  European  trees, 
such  for  instance  as  the  elm  or  the  cherry,  tend  to  have  larger 
leaves  the  further  north  we  go ;  but  in  this  case  the  phenomenon 
is  to  be  ascribed  rather  to  the  longer  hours  of  daylight  than  to 
any  difference  of  temperature*.  The  point  is  a  physiological  one, 
and  consequently  of  little  importance  to  us  here  f ;  the  main  point 
for  the  morphologist  is  the  very  simple  one  that  physical  or 
climatic  conditions  have  greatly  influenced  the  rate  of  growth. 
The  case  is  analogous  to  the  direct  influence  of  temperature  in 
modifying  the  colouration  of  organisms,  such  as  certain  butterflies. 
Now  if  temperature  affects  the  rate  of  growth  in  strict  uniformity, 
alike  in  all  directions  and  in  all  parts  or  organs,  its  direct  effect 
must  be  limited  to  the  production  of  local  races  or  varieties  dift'ering 
from  one  another  in  actual  magnitude,  as  the  Siberian  goldfinch 
or  bullfinch,  for  instance,  differ  from  our  own.  But  if  there  be 
even  ever  so  little  of  a  discriminating  action  in  the  enhancement 
of  growth  by  temperature,  such  that  it  accelerates  the  growth  of 
one  tissue  or  one  organ  more  than  another,  then  it  is  evident  that 
it  must  at  once  lead  to  an  actual  difference  of  racial,  or  even 
"■  specific  "  form. 

It  is  not  to  be  doubted  that  the  various  factors  of  climate 
have  some  such  discriminating  influence.  The  leaves  of  our 
northern  trees  may  themselves  be  an  instance  of  it ;  and  we  have, 

*  It  may  well  be  that  the  effect  is  not  due  to  light  after  all ;  but  to  increased 
absorption  of  heat  by  the  soil,  as  a  result  of  the  long  hours  of  exposure  to  the  sun. 

t  On  growth  in  relation  to  light,  see  Davenport,  Exp.  Morphology,  ii,  ch.  xvii. 
In  some  cases  (as  in  the  roots  of  Peas),  exposure  to  hght  seems  to  have  no  effect 
on  growth;  in  other  cases,  as  in  diatoms  (according  to  Whipple's  experiments, 
quoted  by  Davenport,  n,  p.  423),  the  effect  of  light  on  growth  or  multipHcation 
is  well-marked,  measurable,  and  apparently  capable  of  expression  by  a  logarithmic 
formula.  The  discrepancy  would  seem  to  arise  from  the  fact  that,  while  light- 
energy  always  tends  to  be  absorbed  by  the  chlorophyll  of  the  plant,  converted  into 
chemical  energy,  and  stored  in  the  shape  of  starch  or  other  reserve  materials,  the 
actual  rate  of  growth  depends  on  the  rate  at  which  these  reserves  are  drawn  on: 
and  this  is  another  matter,  in  which  light-energy  is  no  longer  directly  concerned. 


124  THE  RATE   OF   GROWTH  [ch. 

probably,  a  still  better  instance  of  it  in  the  case  of  Alpine  plants  *, 
whose  general  habit  is  dwarfed,  though  their  floral  organs  suffer 
little  or  no  reduction.  The  subject,  however,  has  been  httle 
investigated,  and  great  as  its  theoretic  importance  would  be  to 
us.  we  must  meanwhile  leave  it  alone. 

Osmotic  factors  in  growth. 

The  curves  of  growth  which  we  have  now  been  studying 
represent  phenomena  which  have  at  least  a  two-fold  interest, 
morphological  and  physiological.  To  the  morphologist,  who 
recognises  that  form  is  a  "function"  of  growth,  the  important 
facts  are  mainly  these:  (1)  that  the  rate  of  growth  is  an  orderly 
phenomenon,  with  general  features  common  to  very  various 
organisms,  while  each  particular  organism  has  its  own  character- 
istic phenomena,  or  "specific  constants" ;  (2)  that  rate  of  growth 
varies  with  temperature,  that  is  to  say  with  season  and  with 
climate,  and  with  various  other  physical  factors,  external  and 
internal;  (3)  that  it  varies  in  different  parts  of  the  body,  and 
according  to  various  directions  or  axes ;  such  variations  being 
definitely  correlated  with  one  another,  and  thus  giving  rise  to 
the  characteristic  proportions,  or  form,  of  the  organism,  and  to 
the  changes  in  form  which  it  undergoes  in  the  course  of  its 
development.  But  to  the  physiologist,  the  phenomenon  suggests 
many  other  important  considerations,  and  throws  much  light  on 
the  very  nature  of  growth  itself,  as  a  manifestation  of  chemical 
and  physical  energies. 

To  be  content  to  shew  that  a  certain  rate  of  growth  occurs  in 
a  certain  organism  under  certain  conditions,  or  to  speak  of  the 
phenomenon  as  a  "reaction"  of  the  living  organism  to  its  environ- 
ment or  to  certain  stimuli,  would  be  but  an  example  of  that  "  lack 
of  particularity!"  in  regard  to  the  actual  mechanism  of  physical 
cause  and  effect  with  which  we  are  apt  in  biology  to  be  too  easily 
satisfied.     But  in  the  case  of  rate  of  growth  we  pass  somewhat 

*  Cf.  for  instance,  Nageli's  classical  account  of  the  effect  of  change  of  habitat 
on  Alpine  and  other  plants :    Sitzungsber.  Baier.  Akad.  Wiss.  1865,  pp.  228—284. 

j  Cf.  Blackman,  F.  F.,  Presidential  Address  m  Botany,  Brit.  Ass.  Dublin,  1908. 
The  fact  was  first  enunciated  by  Baudrimont  and  St  Ange,  Recherches  sur  le 
developpement  du  foetus,  Mem.  Acad.  Set.  xi,  p.  469,  1851. 


Ill]  OSMOTIC  FACTORS   IN   GROWTH  125 

beyond  these  limitations ;  for  the  affinity  with  certain  types  of 
chemical  reaction  is  plain,  and  has  been  recognised  by  a  great 
number  of  physiologists. 

A  large  part  of  the  phenomenon  of  growth,  both  in  animals 
and  still  more  conspicuously  in  plants,  is  associated  with  "turgor," 
that  is  to  say,  is  dependent  on  osmotic  conditions ;  in  other  words, 
the  velocity  of  growth  depends  in  great  measure  (as  we  have  already 
seen,  p.  113)  on  the  amount  of  water  taken  up  into  the  living 
cells,  as  well  as  on  the  actual  amount  of  chemical  metabolism 
performed  by  them*.  Of  the  chemical  phenomena  which  result 
in  the  actual  increase  of  protoplasm  we  shall  speak  presently,  but 
the  role  of  water  in  growth  deserves  also  a  passing  word,  even  in 
our  morphological  enquiry. 

It  has  been  shewn  by  Loeb  that  in  Cerianthus  or  Tubularia, 
for  instance,  the  cells  in  order  to  grow  must  be  turgescent;  and 
this  turgescence  is  only  possible  so  long  as  the  salt  water  in  w^hich 
the  cells  he  does  not  overstep  a  certain  Umit  of  concentration.  The 
limit,  in  the  case  of  Tubularia,  is  passed  when  the  salt  amounts 
to  about  5-4  per  cent.  Sea-water  contains  some  3-0  to  3-5  p.c. 
of  salts ;  but  it  is  when  the  sahnity  falls  much  below  this  normal, 
to  about  2-2  p.c.^  that  Tubularia  exhibits  its  maximal  turgescence^ 
and  maximal  growth.  A  further  dilution  is  said  to  act  as  a  poison 
to  the  animal.  Loeb  has  also  shewn  f  that  in  certain  eggs  (e.g. 
those  of  the  little  fish  Fundulus)  an  increasing  concentration  of 
the  sea-water  (leading  to  a  diminishing  "water-content"  of  the 
egg)  retards  the  rate  of  segmentation  and  at  length  renders 
segmentation  impossible;  though  nuclear  division,  by  the  way, 
goes  on  for  some  time  longer. 

Among  many  other  observations  of  the  same  kind,  those  of 
Bialaszewicz  t ,  on  the  early  growth  of  the  frog,  are  notable. 
He  shews  that  the  growth  of  the  embryo  while  still  within  the 

*  Cf.  Loeb,  Untcr.tuchungen  zur  physiol.  Morphologie  der  Thiere,  1892;  also 
Experiments  on  Cleavage,  J.  of  Morph.  vn,  p.  253,  1892;  Zusammenstellung  der 
Ergebnisse  einiger  Arbeiten  iiber  die  Dynamik  des  thierischen  Wachsthum,  Arch, 
f.  Entw.  Mech.  xv,  1902-3,  p.  669 :  Davenport,  On  the  Role  of  Water  in  Growth, 
Bosto7i  Soc.  N.  H.  1897;    Ida  H.  Hyde,  Am.  J.  of  Physiol,  xn,  1905,  p.  241,  etc. 

t  Pfldger's  Archiv,  lv,  1893. 

J  Beitrage  zur  Kenntniss  der  Wachstumsvorgange  bei  Amphibienembryonen, 
B^dl.  Acad.  Sci.  de  Cracovie,  1908,  p.  783 ;  cf.  Arch.  f.  Entw.  Mech.  xxvm,  p.  160, 
1909;  xxxiv,  p.  489,  1912. 


126  THE   RATE   OF   GROWTH  [ch. 

vitelline  membrane  depends  wholly  on  the  absorption  of  water ; 
that  whether  rate  of  growth  be  fast  or  slow  (in  accordance  with 
temperature)  the  quantity  of  water  absorbed  is  constant;  and 
that  successive  changes  of  form  correspond  to  definite  quantities 
of  water  absorbed.  The  solid  residue,  as  Davenport  has  also 
shewn,  may  actually  and  notably  diminish,  while  the  embryo 
organism  is  increasing  rapidly  in  bulk  and  weight. 

On  the  other  hand,  in  later  stages  and  especially  in  the  higher 
animals,  the  percentage  of  water  tends  to  diminish.  This  has 
been  shewn  by  Davenport  in  the  frog,  by  Potts  in  the  chick,  and 
particularly  by  Fehhng  in  the  case  of  man*.  Fehhng's  results 
are  epitomised  as  follows : 

Age  in  weeks         ...       6         17        22        24        26        30        35        39 
Percentage  of  water     97-5     91-8     92-0     89-9     86-4     83-7     82-9     74-2 

And  the  following  illustrate  Davenport's  results  for  the  frog: 


Age  in  weeks 

1 

2 

5 

7 

9 

14 

41 

84 

Percentage  of  water 

56-3 

58-5 

76-7 

89-3 

931 

950 

90-2 

87-5 

To  such  phenomena  of  osmotic  balance  as  the  above,  or  in  other 
words  to  the  dependence  of  growth  on  the  uptake  of  water,  Hober  f 
and  also  Loeb  are  inclined  to  refer  the  modifications  of  form 
which  certain  phyllopod  Crustacea  undergo,  when  the  highly 
saline  waters  which  they  inhabit  are  further  concentrated,  or  are 
abnormally  diluted.  Their  growth,  according  to  Schmankewitsch, 
is  retarded  by  increase  of  concentration,  so  that  the  individuals 
from  the  more  saline  waters  appear  stunted  and  dwarfish;  and 
they  become  altered  or  transformed  in  other  ways,  which  for  the 
most  part  suggest  "degeneration,"  or  a  failure  to  attain  full  and 
perfect  development  J.  Important  physiological  changes  also 
ensue.  The  rate  of  multiplication  is  increased,  and  partheno- 
genetic  reproduction  is  encouraged.  Male  individuals  become 
plentiful  in  the  less  sahne  waters,  and  here  the  females  bring  forth 

*  Febling,  H.,  Arch,  fiir  Gynaekologie,  xi.  1877;  cf.  Morgan,  Experimental 
Zoology,  p.  240,   1907. 

t  Hober,  R.,  Bedeutung  der  Theorie  der  Losungen  fiir  Physiologie  und 
Medizin,  Biol.  Centralhl.  xix,  1899;  cf.  pp.  272-274. 

t  Schmankewitsch  has  made  other  interesting  observations  on  change  of  size 
and  form,  after  some  generations,  in  relation  to  change  of  density;  e.g.  in  the 
flagellate  infusorian  Anisonema  acinus,  Biitschli  {Z.  /.  w.  Z.  xxix,  p.  429,  1877). 


Ill]  OSMOTIC  FACTORS   IN   GROWTH  127 

their  young  alive ;  males  disappear  altogether  in  the  more  con- 
centrated brines,  and  then  the  females  lay  eggs,  which,  however, 
only  begin  to  develop  when  the  salinity  is  somewhat  reduced. 

The  best-known  case  is  the  little  "brine-shrimp,"  Artemia 
salina,  found,  in  one  form  or  another,  all  the  world  over,  and  first 
discovered  more  than  a  century  and  a  half  ago  in  the  salt-pans  at 
Lymington.  Among  many  allied  forms,  one,  A.  ynilhausenii, 
inhabits  the  natron-lakes  of  Egypt  and  Arabia,  where,  under  the 
name  of  "loul,"  or  "Fezzan-worm,"  it  is  eaten  by  the  Arabs*. 
This  fact  is  interesting,  because  it  indicates  (and  investigation 
has  apparently  confirmed)  that  the  tissues  of  the  creature  are  not 
impregnated  with  salt,  as  is  the  medium  in  which  it  lives.  The 
fluids  of  the  body,  the  milieu  interne  (as  Claude  Bernard  called 
them  t),  are  no  more  salt  than  are  those  of  any  ordinary  crus- 
tacean or  other  animal,  but  contain  only  some  0-8  per  cent,  of 
NaClt,  while  the  milieu  externe  may  contain  10,  20,  or  more  per 
cent,  of  this  and  other  salts ;  which  is  as  much  as  to  say  that 
the  skin,  or  body- wall,  of  the  creature  acts  as  a  "semi-permeable 
membrane,"  through  which  the  dissolved  salts  are  not  permitted 
to  diffuse,  though  water  passes  through  freely :  until  a  statical 
equilibrium  (doubtless  of  a  complex  kind)  is  at  length  attained. 

Among  the  structural  changes  which  result  from  increased 
concentration  of  the  brine  (partly  during  the  life-time  of  the 
individual,  but  more  markedly  during  the  short  season  which 
suffices  for  the  development  of  three  or  four,  or  perhaps  more, 
successive  generations),  it  is  found  that  the  tail  comes  to  bear 
fewer  and  fewer  bristles,  and  the  tail-fins  themselves  tend  at  last 
to  disappear;    these  changes  corresponding  to  what  have  been 

*  These  " Fezzan-worms,"  when  first  described,  were  supposed  to  be  "insects' 
eggs";  cf.  Humboldt,  Personal  Narrative,  vi,  i,  8,  note;  Kirby  and  Spence. 
Letter  x. 

t  Cf.  Introd.  a  VeUide  de  la  medecine  experimentile,  1885,  p.  110. 

t  Cf.  Abonyi,  Z.  f.  w.  Z.  cxiv,  p.  134,  1915.  But  Fredericq  has  shewn  that 
the  amount  of  NaCl  in  the  blood  of  Crustacea  (Carcinus  moenas)  varies,  -and 
all  but  corr  s  onds,  with  the  density  of  the  water  in  which  the  creature 
has  been  kept  {Arch,  de  Zool.  Exp.  et  Gen.  (2),  in,  p.  xxxv,  1885);  and 
other  results  of  Fredericq's,  and  various  data  given  or  quoted  by  Bottazzi 
(Osmotischer  Druck  und  elektrische  Leitungsfahigkeit  der  Fliissigkeiten  der 
Organismen,  in  Asher-Spiro's  Ergebn.  d.  Physiologie,  vii,  pp.  160-402,  1908)  suggest 
that  the  case  of  the  brine-shrimps  must  be  looked  upon  as  an  extreme  or  exceptional 
one. 


128 


THE  RATE   OF   GROWTH 


[CH. 


described  as  the  specific  characters  of  A.  milhausenii,  and  of  a 
still  more  extreme  form,  A.  kop'peniana ;  while  on  the  other 
hand,  progressive  dilution  of  the  water  tends  to  precisely  opposite 
conditions,  resulting  in  forms  which  have  also  been  described  as 
separate  species,  and  even  referred  to  a  separate  genus,  Callaonella, 
closely  akin  to  Branchipus  (Fig.  33).  Pari  passu  with  these  changes, 
there  is  a  marked  change  in  the  relative  lengths  of  the  fore  and 
hind  portions  of  the  body,  that  is  to  say,  of  the  "  cephalothorax " 
and  abdomen :  the  latter  growing  relatively  longer,  the  Salter  the 
water.     In  other  words,  not  only  is  the  rate  of  growth  of  the  whole 


^  w  w 


^' 

a 

-j^. 

s 

Ci 

^ 

g5* 

s 

§ 

1 

1 

Co 

2o 

1 

Oc' 

00 

5" 

Artemia  s  sir. 

Callaonella 

Fig.  33.  Brine-shrimps  (Artemia),  from  more  or  less  saline  water.  Upper  figures 
shew  tail-segment  and  tail-fins ;  lower  figures,  relative  length  of  cephalothorax 
and  abdomen.     (After  Abonyi.) 


animal  lessened  by  the  sahne  concentration,  but  the  specific  rates 
of  growth  in  the  parts  of  its  body  are  relatively  changed.  This 
latter  phenomenon  lends  itself  to  numerical  statement,  and  Abonyi 
has  lately  shewn  that  we  may  construct  a  very  regular  curve,  by 
plotting  the  proportionate  length  of  the  creature's  abdomen 
against  the  salinity,  or  density,  of  the  water;  and  the  several 
species  of  Artemia,  with  all  their  other  correlated  specific  characters, 
are  then  found  to  occupy  successive,  more  or  less  well-defined,  and 
more  or  less  extended,  regions  of  the  curve  (Fig.  33).  In  short,  the 
density  of  the  water  is  so  clearly  a   "function"  of  the  specific 


Ill] 


OSMOTIC  FACTORS   IN  GROWTH 


129 


character,  that  we  may  briefly  define  the  species  Artemia  {Callao- 
nella)  Jelskii,  for  instance,  as  the  Artemia  of  density  1000-1010 
(NaCl),  or  the  typical  A.  salina,  or  principalis,  as  the  Artemia 
of  density  1018-1025,  and  so  forth.  It  is  a  most  interesting 
fact  that  these  Artemiae,  under  the  protection  of  their  semi- 
permeable skins,  are  capable  of  living  in  waters  not  only  of 
great  density,  but  of  very  varied  chemical  composition.  The 
natron-lakes,  for  instance,  contain  large  quantities  of  magnesium 


Koppeniana 


160 


140 


^    120 


100 


1000  1020  1040  1060 

Density  of  water 


1080 


2000 


Fig.  34.     Percentage  ratio  of  lengtli  of  abdomen  to  ceplialothorax  in  brine-shrimps, 
at  various  salinities.     (After  Abonyi.) 

sulphate;  and  the  Artemiae  continue  to  live  equally  well  in 
artificial  solutions  where  this  salt,  or  where  calcium  chloride,  has 
largely  taken  the  place  of  sodium  chloride  in  the  more  common 
habitat.  Furthermore,  such  waters  as  those  of  the  natron-lakes 
are  subject  to  very  great  changes  of  chemical  composition  as 
concentration  proceeds,  owing  to  the  different  solubilities  of  the 
constituent  salts.  It  appears  that  the  forms  which  the  Artemiae 
assume,  and  the  changes  which  they  undergo,  are  identical  or 

T.  G.  9 


130  THE   RATE   OF   GROWTH  [ch. 

indistinguishable,  whichever  of  the  above  salts  happen  to  exist, 
or  to  predominate,  in  their  saline  habitat.  At  the  same  time  we 
still  lack  (so  far  as  I  know)  the  simple,  but  crucial  experiments 
which  shall  tell  us  whether,  in  solutions  of  different  chemical 
composition,  it  is  at  equal  densities,  or  at  "isotonic"  concentrations 
(that  is  to  say,  under  conditions  where  the  osmotic  pressure, 
and  consequently  the  rate  of  diffusion,  is  identical),  that  the 
same  structural  changes  are  produced,  or  corresponding  phases 
of  equilibrium  attained. 

While  Hober  and  others*  have  referred  all  these  phenomena  to 
osmosis,  Abonyi  is  inclined  to  believe  that  the  viscosity,  or 
mechanical  resistance,  of  the  fluid  also  reacts  upon  the  organism ; 
and  other  possible  modes  of  operation  have  been  suggested. 
But  we  may  take  it  for  certain  that  the  phenomenon  as  a  whole 
is  not  a  simple  one;  and  that  it  includes  besides  the  passive 
phenomena  of  intermolecular  diffusion,  some  other  form  of  activity 
wliich  plays  the  part  of  a  regulatory  mechanismf . 

Growth  and  catalytic  action. 

In  ordinary  chemical  reactions  we  have  to  deal  (1)  with  a 
specific  velocity  proper  to  the  particular  reaction,  (2)  with  varia- 
tions due  to  temperature  and  other  physical  conditions,  (3)  according 
to  van't  Hoff's  "  Law  of  Mass,"  with  variations  due  to  the  actual 
quantities  present  of  the  reacting  substances,  and  (4)  in  certain 
cases,  with  variations  due  to  the  presence  of  "catalysing  agents." 
In  the  simpler  reactions,  the  law  of  mass  involves  a  steady,  gradual 
slowing-down  of  the  process,  according  to  a  logarithmic  ratio,  as 
the  reaction  proceeds  and  as  the  initial  amount  of  substance 
diminishes ;    a  phenomenon,  however,  which  need  not  necessarily 

*  Cf.  Sehmankewitsch,  Z.  f.  w.  Zool.  xxv,  1875,  xxix,  1877,  etc. ;  transl.  in 
appendix  to  Packard's  Monogr.  of  N.  American  Phyllopoda,  1883,  pp.  466-514 
Daday  de  Dees,  Ann.  Sci.  Nat.  (Zool.),  (9),  xi,  1910;  Samter  imd  Heymons,  Abh 
d.  K.  pr.  Akad.  Wiss.  1902;  Bateson,  Mat.  for  the  Study  of  Variation,  1894,  pp 
96-101;  Anikin,  Mitth.  Kais.  Univ.  Tomsk,  xiv:  Zool.  Ce7itralM.  vi,  pp.  756-760 
1908;   Abonyi,  Z.f.  w.  Z.  cxiv,  pp.  96-168,  1915  (with  copious  bibliograpliy),  etc 

t  According  to  the  empirical  canon  of  physiology,  that  (as  Fredericq  expresses 
it)  "L'etre  vivant  est  agence  de  telle  maniere  que  chaque  influence  pertui-batrice 
provoque  d'elle-meme  la  mise  en  activite  de  I'appareil  compensateur  qui  doit 
neutrahser  et  reparer  le  dommage." 


Ill]  GROWTH  AND   CATALYTIC  ACTION  131 

occur  in  the  organism,  part  of  whose  energies  are  devoted  to  the 
continual  bringing-up  of  fresh  supplies. 

Catalytic  action  occurs  when  some  substance,  often  in  very 
minute  quantity,  is  present,  and  by  its  presence  produces  or 
accelerates  an  action,  by  opening  "  a  way  round,"  without 
the  catalytic  agent  itself  being  diminished  or  used  up*. 
Here  the  velocity  curve,  though  quickened,  is  not  necessarily 
altered  in  form,  for  gradually  the  law  of  mass  exerts  its 
effect  and  the  rate  of  the  reaction  gradually  diminishes.  But 
in  certain  cases  we  have  the  very  remarkable  phenomenon  that 
a  body  acting  as  a  catalyser  is  necessarily  formed  as  a  product, 
or  bye-product,  of  the  main  reaction,  and  in  such  a  case  as  this 
the  reaction-velocity  will  tend  to  be  steadily  accelerated.  Instead 
of  dwind  ing  away,  the  reaction  will  continue  with  an  ever- 
increasing  velocity :  always  subject  to  the  reservation  that  limiting 
conditions  will  in  time  make  themselves  felt,  such  as  a  failure  of 
some  necessary  ingredient,  or  a  development  of  some  substance 
which  shall  antagonise  or  finally  destroy  the  original  reaction. 
Such  an  action  as  this  we  have  learned,  from  Ostwald,  to  describe 
as  "autocatalysis."  Now  we  know  that  certain  products  of 
protoplasmic  metabolism,  such  as  the  enzymes,  are  very  powerful 
catalysers,  and  we  are  entitled  to  speak  of  an  autocatalytic  action 
on  the  part  of  protoplasm  itself.  This  catalytic  activity  of  pro- 
toplasm is  a  very  important  phenomenon.  As  Blackman  says, 
in  the  address  already  quoted,  the  botanists  (or  the  zoologists) 
"call  it  groivth,  attribute  it  to  a  specific  power  of  protoplasm  for 
assimilation,  and  leave  it  alone  as  a  fundamental  phenomenon; 
but  they  are  much  concerned  as  to  the  distribution  of  new  growth 
in  innumerable  specifically  distinct  forms."  While  the  chemist,  on 
the  other  hand,  recognises  it  as  a  familiar  phenomenon,  and  refers  it 
to  the  same  category  as  his  other  known  examples  of  autocatalysis. 

*  Such  phenomena  come  precisely  under  the  head  of  what  Bacon  called 
Instances  of  Magic :  '  By  which  I  mean  those  wherein  the  material  or  efficient 
cause  is  scanty  and  small  as  compared  with  the  work  or  effect  produced ;  so  that 
even  when  they  are  common,  they  seem  like  miracles,  some  at  first  iight,  others 
even  after  attentive  consideration.  These  magical  effects  are  brought  about  in 
three  ways... [of  which  one  isj  by  excitation  or  invitation  in  another  body,  as  in 
the  magnet  which  excites  numberless  needles  without  losing  any  of  its  virtue,  or 
in  yeast  and  such-like."     Nov.  Org.,  cap.  li. 

9—2 


132  THE   RATE   OF   GROWTH  [ch. 

This  very  important,  and  perhaps  even  fundamental  pheno- 
menon of  growth  would  seem  to  have  been  first  recognised  by 
Professor  Chodat  of  Geneva,  as  we  are  told  by  his  pupil  Monnier  *. 
"On  peut  bien,  ainsi  que  M.  Chodat  I'a  propose,  considerer 
I'accroissement  comme  une  reaction  chimique  complexe,  dans 
laquelle  le  catalysateur  est  la  cellule  vivante,  et  les  corps  en 
presence  sont  Teau,  les  sels,  et  I'acide  carbonique." 

Very  soon  afterwards  a  similar  suggestion  was  made  by  Loebt, 
in  connection  with  the  synthesis  of  nuclein  or  nuclear  protoplasm ; 
for  he  remarked  that,  as  in  an  autocatalysed  chemical  reaction, 
the  velocity  of  the  synthesis  increases  during  the  initial  stage  of 
cell-division  in  proportion  to  the  amount  of  nuclear  matter  already 
synthesised.  In  other  words,  one  of  the  products  of  the  reaction, 
i.e.  one  of  the  constituents  of  the  nucleus,  accelerates  the  pro- 
duction of  nuclear  from  cytoplasmic  material. 

The  phenomenon  of  autocatalysis  is  by  no  means  confined  to 
living  or  protoplasmic  chemistry,  but  at  the  same  time  it  is 
characteristically,  and  apparently  constantly,  associated  therewith. 
And  it  would  seem  that  to  it  we  may  ascribe  a  considerable  part 
of  the  difference  between  the  growth  of  the  organism  and  the 
simpler  growth  of  the  crystal  J  :  the  fact,  for  instance,  that  the  cell 
can  grow  in  a  very  low  concentration  of  its  nutritive  solution, 
while  the  crystal  grows  only  in  a  supersaturated  one ;  and  the 
fundamental  fact  that  the  nutritive  solution  need  only  contain 
the  more  or  less  raw  materials  of  the  complex  constituents  of  the 
cell,  while  the  crystal  grows  only  in  a  solution  of  its  own  actual 
substance  § . 

As  F.  F.  Blackman  has  pointed  out,  the  multiplication  of  an 
organism,  for  instance  the  prodigiously  rapid  increase  of  a  bacterium, 

*  Monnier,  A.,  Les  matieres  minerales,  et  la  loi  d'accroissement  des  Vegetaux, 
Publ.  de  Vlnst.  de  Bot.  de  VUniv.  de,  Geneve  (7),  in,  1905.  Cf.  Robertson,  On  the 
Normal  Rate  of  Growth  of  an  Individual,  and  its  Biochemical  Significance,  Arch, 
f.  Entw.  Mech.  xxv,  pp.  581-614,  xxvi,  pp.  108-118,  1908;  Wolfgang  Ostwald, 
Die  zeitlichen  Eigenschafien  der  Eniwickelungsvonjdnge,  1908;  Hatai,  S.,  Interpreta- 
tion of  Growth-curves  from  a  Dynamical  Standpoint,  Anat.  Record,  v,  p.  373, 
1911. 

t  Biochem.  Zeitschr.  n.  1906,  p.  34. 

J  Even  a  crystal  may  be  said,  in  a  sense,  to  display  "autocatalysis":  for  the 
bigger  its  surface  becomes,  the  more  rapidly  does  the  mass  go  on  increasing. 

§   Cf.  Loeb,  The  Stimulation  of  Growth,  Science,  May  14,  1915. 


Ill]  GROWTH   AND   CATALYTIC  ACTION  133 

which  tends  to  double  its  numbers  every  few  minutes,  till  (were 
it  not  for  limiting  factors)  its  numbers  would  be  all  but  incalculable 
in  a  day*,  is  a  simple  but  most  striking  illustration  of  the  potenti- 
alities of  protoplasmic  catalysis ;  and  (apart  from  the  large  share 
taken  by  mere  "turgescence"'  or  imbibition  of  water)  the  same 
is  true  of  the  growth,  or  cell-multiplication,  of  a  multicellular 
organism  in  its  first  stage  of  rapid  acceleration. 

It  is  not  necessary  for  us  to  pursue  this  subject  much  further, 
for  it  is  sufficiently  clear  that  the  normal  "curve  of  growth"  of 
an  organism,  in  all  its  general  features,  very  closely  resembles  the 
velocity-curve  of  chemical  autocatalysis.  We  see  in  it  the  first 
and  most  typical  phase  of  greater  and  greater  acceleration ;  this 
is  followed  by  a  phase  in  which  limiting  conditions  (whose  details 
are  practically  unknown)  lead  to  a  falling  off  of  the  former 
acceleration ;  and  in  most  cases  we  come  at  length  to  a  third  phase, 
in  which  retardation  of  growth  is  succeeded  by  actual  diminution 
of  mass.  Here  we  may  recognise  the  influence  of  processes,  or 
of  products,  which  have  become  actually  deleterious ;  their 
deleterious  influence  is  staved  off  for  awhile,  as  the  organism  draws 
on  its  accumulated  reserves,  but  they  lead  ere  long  to  the  stoppage 
of  all  activity,  and  to  the  physical  phenomenon  of  death.  But 
when  we  have  once  admitted  that  the  limiting  conditions  of 
growth,  which  cause  a  phase  of  retardation  to  follow  a  phase 
of  acceleration,  are  very  imperfectly  known,  it  is  plain  that, 
ipso  facto,  we  must  admit  that  a  resemblance  rather  than  an 
identity  between  this  phenomenon  and  that  of  chemical  auto- 
catalysis is  all  that  we  can  safely  assert  meanwhile.  Indeed,  as 
Enriques  has  shewn,  points  of  contrast  between  the  two  phenomena 
are  not  lacking ;  for  instance,  as  the  chemical  reaction  draws  to 
a  close,  it  is  by  the  gradual  attainment  of  chemical  equilibrium: 
but  when  organic  growth  draws  to  a  close,  it  is  by  reason  of  a  very 
different  kind  of  equilibrium,  due  in  the  main  to  the  gradual 
differentiation  of  the  organism  into  parts,  among  whose  peculiar 

*  B.  coli-comynunis,  according  to  Buchner,  tends  to  double  in  22  minutes;  in 
24  hours,  therefore,  a  single  individual  would  be  multiplied  by  something  like 
10-8;  Sitzung.<<ber.  Munchen.  Ges.  MorphoJ.  u.  Physiol,  in,  pp.  65-71,  1888.  Cf. 
Marshall  Ward,  Biology  of  Bacillus  ramosus,  etc.  Pr.  R.  S.  lviii,  265-468,  1895. 
The  comparatively  large  infusorian  Stylonichia,  according  to  Maupas,  would 
multiply  in  a  month  by  10*^. 


134  THE  KATE   OF   GROWTH  [ch. 

and  specialised  functions  that  of  cell-multiplication  tends  to  fall 
into  abeyance*. 

It  would  seem  to  follow,  as  a  natural  consequence,  from  what 
has  been  said,  that  we  could  without  much  difficulty  reduce  our 
curves  of  growth  to  logarithmic  formulae  |  akin  to  those  which 
the  physical  chemist  finds  apphcable  to  his  autocatalytic  reactions. 
This  has  been  diligently  attempted  by  various  writersj ;  but  the 
results,  while  not  destructive  of  the  hypothesis  itself,  are  only 
partially  successful.  The  difficulty  arises  mainly  from  the  fact 
that,  in  the  life-history  of  an  organism,  we  have  usually  to  deal 
(as  indeed  we  have  seen)  with  several  recurrent  periods  of  relative 
acceleration  and  retardation.  It  is  easy  to  find  a  formula  which 
shall  satisfy  the  conditions  during  any  one  of  these  periodic 
phases,  but  it  is  very  difficult  to  frame  a  comprehensive  formula 
which  shall  apply  to  the  entire  period  of  growth,  or  to  the  whole 
duration  of  fife. 

But  if  it  be  meanwhile  impossible  to  formulate  or  to  solve  in 
precise  mathematical  terms  the  equation  to  the  growth  of  an 
organism,  we  have  yet  gone  a  very  long  way  towards  the  solution 
of  such  problems  when  we  have  found  a  "qualitative  expression," 
as  Poincare  puts  it;  that  is  to  say,  when  we  have  gained  a  fair 
approximate  knowledge  of  the  general  curve  which  represents  the 
unknown  function. 

As  soon  as  we  have  touched  on  such  matters  as  the  chemical 
phenomenon  of  catalysis,  we  are  on  the  threshold  of  a  subject 
which,  if  we  were  able  to  pursue  it,  would  soon  lead  us  far  into 
the  special  domain  of  physiology ;  and  there  it  would  be  necessary 
to  follow  it  if  we  were  dealing  with  growth  as  a  phenomenon  in 
itself,  instead  of  merely  as  a  help  to  our  study  and  comprehension 
of  form.  For  instance  the  whole  question  of  diet,  of  overfeeding 
and  underfeeding,  would  present  itself  for  discussion  §.  But 
without  attempting  to  open  up  this  large  subject,  we  may  say  a 

*  Cf.  Enriques,  Wachsthum  und  seine  analytisohe  Darstellung,  Biol.  Centralbl. 
1909,  p.  337. 

f  Cf.  (int.  al.)  Mellor,  Chemical  Statics  and  Dynamics,  1904,  p.  291. 

t   Cf.  Robertson,  I.e. 

§  See,  for  a  brief  resume  of  this  subject,  Morgan's  Experimental  Zoology, 
chap.  xvi. 


Ill]  GROWTH  AND   CATALYTIC  ACTION  135 

further  passing  word  upon  the  essential  fact  that  certain  chemical 
substances  have  the  power  of  accelerating  or  of  retarding,  or  in 
some  way  regulating,  growth,  and  of  so  influencing  directly  the 
morphological  features  of  the  organism. 

Thus  lecithin  has  been  shewn  by  Hatai*,  Danilewskyf  and 
others  to  have  a  remarkable  power  of  stimulating  growth  in 
various  animals;  and  the  so-called  "auximones,"  which  Professor 
Bottomley  prepares  by  the  action  of  bacteria  upon  peat  appear 
to  be,  after  a  somewhat  similar  fashion,  potent  accelerators  of 
the  growth  of  plants.  But  by  much  the  most  interesting  cases, 
from  our  point  of  view,  are  those  where  a  particular  substance 
appears  to  exert  a  differential  effect,  stimulating  the  growth  of 
one  part  or  organ  of  the  body  more  than  another. 

It  has  been  known  for  a  number  of  years  that  a  diseased 
condition  of  the  pituitary  body  accompanies  the  phenomenon 
known  as  "acromegaly,"  in  which  the  bones  are  variously  enlarged 
or  elongated,  and  w^hich  is  more  or  less  exemplified  in  every 
skeleton  of  a  '"giant"  ;  while  on  the  other  hand,  disease  or  extirpa- 
tion of  the  thyroid  causes  an  arrest  of  skeletal  development,  and, 
if  it  take  place  early,  the  subject  remains  a  dwarf.  These,  then, 
are  well-known  illustrations  of  the  regulation  of  function  by  some 
internal  glandular  secretion,  some  enzyme  or  "hormone"  (as 
Bayhss  and  Starling  call  it),  or  "harmozone,"  as  Gley  calls  it  in 
the  particular  case  where  the  function  regulated  is  that  of  growth, 
with  its  consequent  influence  on  form. 

Among  other  illustrations  (which  are  plentiful)  we  have,  for 
instance  the  growth  of  the  placental  decidua,  which  Loeb  has 
shewn  to  be  due  to  a  substance  given  off  by  the  corpus  luteum 
of  the  ovary,  giving  to  the  uterine  tissues  an  abnormal  capacity 
for  growth,  which  in  turn  is  called  into  action  by  the  contact  of 
the  ovum,  or  even  of  any  foreign  body.  And  various  sexual 
characters,  such  as  the  plumage,  comb  and  spurs  of  the  cock, 
are  believed  in  like  manner  to  arise  in  response  to  some  particular 
internal  secretion.  When  the  source  of  such  a  secretion  is  removed 
by  castration,  well-known  morphological  changes  take  place  in 
various  animals;  and  when  a  converse  change  takes  place,  the 
female  acquires,  in  greater  or  less  degree,  characters  which  are 

*  Amer.  J.  of  Physiol,  x,  1904.  f  C'.J?.  cxxi,  cxxii,  1895-96. 


136  THE   RATE   OF   GROWTH  [ch. 

proper  to  the  male,  as  in  certain  extreme  cases,  known  from  time 
immemorial,  when  late  in  life  a  hen  assumes  the  plumage  of  the 
cock. 

There  are  some  very  remarkable  experiments  by  Gudernatsch, 
in  which  he  has  shewn  that  by  feeding  tadpoles  (whether  of  frogs 
or  toads)  on  thyroid  gland  substance,  their  legs  may  be  made  to 
grow  out  at  any  time,  days  or  weeks  before  the  normal  date  of 
their  appearance*.  No  other  organic  food  was  found  to  produce 
the  same  effect ;  but  since  the  thyroid  gland  is  known  to  contain 
iodine  f ,  Morse  experimented  with  this  latter  substance,  and  found 
that  if  the  tadpoles  were  fed  with  iodised  amino-acids  the  legs 
developed  precociously,  just  as  when  the  thyroid  gland  itself  was 
used.  We  may  take  it,  then,  as  an  established  fact,  whose  full 
extent  and  bearings  are  still  awaiting  investigation,  that  there 
exist  substances  both  within  and  without  the  organism  which 
have  a  marvellous  power  of  accelerating  growth,  and  of  doing  so 
in  such  a  way  as  to  affect  not  only  the  size  but  the  form  or  pro- 
portions of  the  organism. 

If  we  once  admit,  as  we  are  now  bound  to  do,  the  existence 
of  such  factors  as  these,  which,  by  their  physiological  activity 
and  apart  from  any  direct  action  of  the  nervous  system,  tend 
towards  the  acceleration  of  growth  and  consequent  modification 
of  form,  we  are  led  into  wide  fields  of  speculation  by  an  easy  and 
a  legitimate  pathway.  Professor  Gley  carries  such  speculations 
a  long,  long  way :  for  he  says  J  that  by  these  chemical  influences 
"Toute  une  partie  de  la  construction  des  etres  parait  s'expliquer 
d'une  fa^on  toute  mecanique.  La  forteresse,  si  longtemps  inacces- 
sible, du  vitalisme  est  entamee.  Car  la  notion  morphogenique 
etait,  suivant  le  mot  de  Dastre§,  comme  'le  dernier  reduit  de  la 
force  vitale.' " 

The  physiological  speculations  we  need  not  discuss :  but,  to 
take  a  single  example  from  morphology,  we  begin  to  understand 
the  possibility,  and  to  comprehend  the  probable  meaning,  of  the 

*  Cf.  Loeb,  Science,  May  14,  191.5, 

f  Cf.  Baumann  u.  Roos,  Vorkommen  von  lod  im  Thierkorper,  Zeitschr.  fur 
Physiol.  Chem.  xxi,  xxii,  1895,  6. 

%  Le  Neo- Vitalisme,  Rev.  Scientifique,  Mars  1911,  p.  22  (of  reprint). 
§  La  vie  et  la  mort,  p.  43,  1902. 


Ill]  GEOWTH  AND   CATALYTIC   ACTION  137 

all  but  sudden  appearance  on  the  earth  of  such  exaggerated  and 
almost  monstrous  forms  as  those  of  the  great  secondary  reptiles 
and  the  great  tertiary  mammals*.  We  begin  to  see  that  it  is  in 
order  to  account,  not  for  the  appearance,  but  for  the  disappearance 
of  such  forms  as  these  that  natural  selection  must  be  invoked. 
And  we  then,  I  think,  draw  near  to  the  conclusion  that  what  is 
true  of  these  is  universally  true,  and  that  the  great  function  of 
natural  selection  is  not  to  originate,  but  to  remove :  donee  ad 
inter itum  genus  id  natura  redegitf. 

The  world  of  things  living,  like  the  world  of  things  inanimate, 
grows  of  itself,  and  pursues  its  ceaseless  course  of  creative  evolution. 
It  has  room,  wide  but  not  unbounded,  for  variety  of  living  form 
and  structure,  as  these  tend  towards  their  seemingly  endless,  but 
yet  strictly  limited,  possibilities  of  permutation  and  degree :  it 
has  room  for  the  great  and  for  the  small,  room  for  the  weak  and 
for  the  strong;  Environment  and  circumstance  do  not  always 
make  a  prison,  wherein  perforce  the  organism  must  either  live 
or  die ;  for  the  ways  of  life  may  be  changed,  and  many  a  refuge 
found,  before  the  sentence  of  unfitness  is  pronounced  and  the 
penalty  of  extermination  paid.  But  there  comes  a  time  when 
"variation,"  in  form,  dimensions,  or  other  qualities  of  the  organism, 
goes  farther  than  is  compatible  with  all  the  means  at  hand  of 
health  and  welfare  for  the  individual  and  the  stock ;  when,  under 
the  active  and  creative  stimulus  of  forces  from  within  and  from 
without,  the  active  and  creative  energies  of  growth  pass  the 
bounds  of  physical  and  physiological  equilibrium :  and  so  reach 
the  limits  which,  as  again  Lucretius  tells  us,  natural  law  has  set 
between  what  may  and  what  may  not  be, 

"et  quid  quaeque  queant  per  foedera  naturai 
quid  porro  nequeant." 
Then,  at  last,  we  are  entitled  to  use  the  customary  metaphor, 
and  to  see  in  natural  selection  an  inexorable  force,  whose  function 

*  Cf.  Dendy,  Evolutionary  Biology,  1912,  p.  408;  Brit.  Ass.  Report  (Portsmouth), 
1911,  p.  278. 

I  Lucret.  v,  877.  "Lucretius  nowhere  seems  to  recognise  the  possibility  of 
improvement  or  change  of  species  by  'natural  selection';  the  animals  remain  as 
they  were  at  the  first,  except  that  the  weaker  and  more  useless  kinds  have  been 
crushed  out.  Hence  he  stands  in  marked  contrast  with  modern  evolutionists." 
Kelsey's  note,  ad  loc. 


138  THE   RATE   OF   GROWTH  [cs. 

is  not  to  create  but  to  destroy, — to  weed,  to  prune,  to  cut  down 
and  to  cast  into  the  fire*. 


Regeneration,  or  growth  and  repair. 

The  phenomenon  of  regeneration,  or  the  restoration  of  lost  or 
amputated  parts,  is  a  particular  case  of  growth  which  deserves 
separate  consideration..  As  we  are  all  aware,  this  property  is 
manifested  in  a  high  degree  among  invertebrates  and  many  cold- 
blooded vertebrates,  diminishing  as  we  ascend  the  scale,  until  at 
length,  in  the  warm-blooded  animals,  it  lessens  down  to  no  more 
than  that  vis  niedicatrix  which  heals  a  wound.  Ever  since  the 
days  of  Aristotle,  and  especially  since  the  experiments  of  Trembley, 
Reaumur  and  Spallanzani  in  the  middle  of  the  eighteenth  century, 
the  physiologist  and  the  psychologist  have  ahke  recognised  that 
the  phenomenon  is  both  perplexing  and  important.  The  general 
phenomenon  is  amply  discussed  elsewhere,  and  we  need,  only 
deal  with  it  in  its  immediate  relation  to  growth  f. 

Regeneration,  hke  growth  in  other  cases,  proceeds  with  a 
velocity  which  varies  according  to  a  definite  law ;  the  rate  varies 
with  the  time,  and  we  may  study  it  as  velocity  and  as  acceleration. 

Let  us  take,  as  an  instance,  Miss  M.  L.  Durbin's  measurements 
of  the  rate  of  regeneration  of  tadpoles'  tails :  the  rate  being  here 
measured  in  terms,  not  of  mass,  but  of  length,  or  longitudinal 
increment  t. 

From  a  number  of  tadpoles,  whose  average  length  was  34-2  mm., 
their  tails  being  on  an  average  21-2  mm.  long,  about  half  the  tail 

*  Even  after  we  have  so  narrowed  the  scope  and  sphere  of  natural  selection, 
it  is  still  hard  to  understand ;  for  the  causes  of  extinction  are  often  wellnigh  as  hard 
to  comprehend  as  are  those  of  the  origin  of  species.  If  we  assert  (as  has  been 
lightly  done)  that  Smilodon  perished  owing  to  its  gigantic  tusks,  that  Teleosaurus 
was  handicapped  by  its  exaggerated  snout,  or  Stegosaurus  weighed  down  by  its 
intolerable  load  of  armour,  we  may  be  reminded  of  other  kindred  forms  to  show 
that  similar  conditions  did  not  necessarily  lead  to  extermination,  or  that  rapid 
extinction  ensued  apart  from  any  such  visible  or  apparent  disadvantages.  Cf. 
Lucas,  F.  A.,  On  Momentum  in  Variation,  Amer.  Nat.  xh,  p.  46,  1907. 

t  See  Professor  T.  H.  Morgan's  Regeneration  (316  pp.),  1901  for  a  full  account 
and  copious  bibliography.  The  early  experiments  on  regeneration,  by  Vallisneri, 
Reaumur,  Bonnet,  Trembley,  Baster,  and  others,  are  epitomised  by  HaUer,  Elem. 
Physiologiae,  vm,  p.  156  seq. 

J  Journ.  Experim.  Zool.  vii,  p.  ,397,  1909. 


3 

7 

10 

14 

18 

24 

30 

L-4 

3-4 

4-3 

5-2 

5-5 

6-2 

6-5 

^•4 

20 

0-9 

0-9 

0-3 

0-7 

0-3 

III]     REGENERATION,   OR   GROWTH  AND   REPAIR    139 

(11-5  mm.)  was  cut  o&,  and  the  amounts  regenerated  in  successive 
periods  are  shewn  as  follows : 

Days  after  operation 

(1)  Amount  regenerated  in  mm. 

(2)  Increment  during  each  period 
(3)(?)     Rate     per     day     during 

each  period  0-46     0-50    0-30    0-25     0-07     0-12     0-05 

The  first  line  of  numbers  in  this  table,  if  plotted  as  a  curve 
against  the  number  of  days,  will  give  us  a  very  satisfactory  view 
of  the  "curve  of  growth"  within  the  period  of  the  observations: 
that  is  to  say,  of  the  successive  relations  of  length  to  time,  or  the 
velocity  of  the  process.  But  the  third  line  is  not  so  satisfactory, 
and  must  not  be  plotted  directly  as  an  acceleration  curve.  For 
it  is  evident  that  the  "rates"  here  determined  do  not  correspond 
to  velocities  at  the  dates  to  which  they  are  referred,  but  are  the 
mean  velocities  over  a  preceding  period ;  and  moreover  the  periods 
over  which  these  means  are  taken  are  here  of  very  unequal  length. 
But  we  may  draw  a  good  deal  more  information  from  this  experi- 
ment, if  we  begin  by  drawing  a  smooth  curve,  as  nearly  as  possible 
through  the  points  corresponding  to  the  amounts  regenerated 
(according  to  the  first  line  of  the  table) ;  and  if  we  then  interpolate 
from  this  smooth  curve  the  actual  lengths  attained,  day  by 
day,  and  derive  from  these,  by  subtraction,  the  successive  daily 
increments,  which  are  the  measure  of  the  daily  mean  velocities 
(Table,  p.  141).  (The  more  accurate  and  strictly  correct  method 
would  be  to  draw  successive  tangents  to  the  curve.) 

In  our  curve  of  growth  (Fig.  35)  we  cannot  safely  interpolate 
values  for  the  first  three  days,  that  is  to  say  for  the  dates  between 
amputation  and  the  first  actual  measurement  of  the  regenerated 
part.  What  goes  on  in  these  three  days  is  very  important;  but 
we  know  nothing  about  it,  save  that  our  curve  descended  to  zero 
somewhere  or  other  within  that  period.  As  we  have  already 
learned,  we  can  more  or  less  safely  interpolate  between  known 
points,  or  actual  observations;  but  here  we  have  no  known 
starting-point.  In  short,  for  all  that  the  observations  tell  us, 
and  for  all  that  the  appearance  of  the  curve  can  suggest,  the 
curve  of  growth  may  have  descended  evenly  to  the  base-hne, 
which  it  would  then  have  reached  about  the  end  of  the  second 


140 


THE  RATE   OF   GROWTH 


[CH. 


day ;    or  it  may  have  had  within  the  first  three  days  a  change  of 
direction,  or  "point  of  inflection,"  and  may  then  have  sprung 


0       2      4      6      8     10    12    14   16    18    20   22    24    26   28  30 

days 

Fig.  35.     Curve  of  regenerative  gro\\i;h  in  tadpoles'  tails.     (From 
M.  L.  Durbin's  data.) 

at  once  from  the  base-hne  at  zero.     That  is  to  say,  there  may 
have  been  an  intervening  "latent  period,"  during  which  no  growth 


2  4   6   8  10  12  14  16  18  20  22  24  26  28  30 

days 

Fig.  3H.     Mean  daily  increments,  corresponding  to  Fig.  3.5. 


Ill]     REGENERATION,   OR   GROWTH  AND   REPAIR    141 

occurred,  between  the  time  of  injury  and  the  first  measurement 
of  regenerative  growth ;  or,  for  all  we  yet  know,  regeneration 
may  have  begun  at  once,  but  with  a  velocity  much  less  than  that 
which  it  afterwards  attained.  This  apparently  trifling  difference 
would  correspond  to  a  very  great  difference  in  the  nature  of  the 
phenomenon,  and  would  lead  to  a  very  striking  difference  in  the 
curve  which  we  have  next  to  draw. 

The  curve  already  drawn  (Fig.  35)  illustrates,  as  we  have  seen, 
the  relation  of  length  to  time,  i.e.  LjT  =  F.  The  second  (Fig.  36) 
represents  the  rate  of  change  of  velocity;    it  sets  F  against  T ; 


oing 

table,  extended  by  graphic 

interfolation 

Total               Daily 

Days 
1 
2 

increment       inci 

ement      Logs  of  do. 



— 

— 

3 

1-40 

60 

1-78 

4 

2-00 

52 

1-72 

5 

2-52 

45 

b65 

6 

2-97 

43 

1-63 

7 

3-40 

32 

1-51 

8 

3-72 

30 

1-48 

9 

4-02 

28 

1-45 

10 

4-30 

22 

1-34 

11 

4-52 

21 

1-32 

12 

4-73 

19 

1-28 

13 

4-92 

18 

1-26 

14 

5-10 

17 

1-23 

15 

5-27 

13 

Ml 

16 

5-40 

14 

M5 

17 

5-54 

13 

Ml 

18 

5-67 

•11 

104 

19 

5-78 

10 

100 

20 

5-88 

10 

1-00 

21 

5-98 

09 

•95 

22 

6-07 

07 

•85 

23 

614 

•07 

•84 

24 

6-21 

•08 

•90 

25 

6-29 

■06 

•78 

26 

6-35 

•06 

•78 

27 

6-41 

•05 

•70 

28 

6-46 

04 

•60 

29 

6-50 

•03 

•48 

30 

6-53 

142 


THE   KATE   OF   GROWTH 


[CH. 


and  V/T  or  L/T- ,  represents  (as  we  have  learned)  the  acceleration  of 
growth,  this  being  simply  the  "differential  coejficient,"  the  first 
derivative  of  the  former  curve. 

Now,  plotting  this  acceleration  curve  from  the  date  of  the 
first  measurement  made  three  days  after  the  amputation  of  the 
tail  (Fig.  36),  we  see  that  it  has  no  point  of  inflection,  but  falls 
steadily,  only  more  and  more  slowly,  till  at  last  it  comes  down 
nearly  to  the  base-line.  The  velocities  of  growth  are  continually 
diminishing.  As  regards  the  missing  portion  at  the  beginning  of 
the  curve,  we  cannot  be  sure  whether  it  bent  round  and  came  dowii 


1-8 
logs 

\- 

1-6 

\. 

1-4 

\^^ 

1-2 

x^^ 

1-0 

X 

•8 

X,-       - 

•6 

\ 

•4 

i               .                1               ;               1                1               r               1                1               1               1               1               • 

2   4   6   8  10  12  14  16  18  20  22  24  26  28  30 

da^s 

Fig.  37.     Logarithms  of  values  shewn  in  Fig.  36. 


to  zero,  or  whether,  as  in  our  ordinary  acceleration  curves  of  growth 
from  birth  onwards,  it  started  from  a  maximum.  The  former  is, 
in  this  case,  obviously  the  more  probable,  but  we  cannot  be  sure. 
As  regards  that  large  portion  of  the  curve  which  we  are 
acquainted  with,  we  see  that  it  resembles  the  curve  known  as 
a  rectangular  hyperbola,  which  is  the  form  assumed  when  two 
A''ariables  (in  this  case  V  and  T)  vary  inversely  as  one  another. 
If  we  take  the  logarithms  of  the  velocities  (as  given  in  the  table) 
and  plot  them  against  time  (Fig.  37),  we  see  that  they  fall,  approxi- 
mately, into  a  straight  line;    and  if  this  curve  be  plotted  on  the 


m]     REGENERATION,   OR   GROWTH  AND   REPAIR    143 

proper  scale  we  shall  find  that  the  angle  which  it  makes  with  the 
base  is  about  25°,  of  which  the  tangent  is  46,  or  in  round  numbers  |. 
Had  the  angle  been  45°  (tan  45°  =  1),  the  curve  would  have 
been  actually  a  rectangular  hyperbola,  with  VT  =  constant.  As 
it  is,  we  may  assume,  provisionally,  that  it  belongs  to  the  same 
family  of  curves,  so  that  F'^T'S  or  V^'^'T,  or  VT"'"^;  are  all  severally 
constant.  In  other  words,  the  velocity  varies  inversely  as  some 
power  of  the  time,  or  vice  versa.  And  in  this  particular  case,  the 
equation  VT^  =  constant,  holds  very  nearly  true ;  that  is  to  say 
the  velocity  varies,  or  tends  to  vary,  inversely  as  the  square  of 


16       1i 
days 

Fig.  38.     Rate  of  regenerative  growth  in  larger  tadpoles. 

the  time.  If  some  general  law  akin  to  this  could  be  established 
as  a  general  law,  or  even  as  a  common  rule,  it  would  be  of  great 
importance. 

But  though  neither  in  this  case  nor  in  any  other  can  the  minute 
increments  of  growth  during  the  first  few  hours,  or  the  first  couple 
of  days,  after  injury,  be  directly  measured,  yet  the  most  important 
point  is  quite  capable  of  solution.  What  the  foregoing  curve 
leaves  us  in  ignorance  of,  is  simply  whether  growth  starts  at  zero, 
with  zero  velocity,  and  works  up  quickly  to  a  maximum  velocity 
from  which  it  afterwards  gradually  falls  away;  or  whether  after 
a  latent  period,  it  begins,  so  to  speak,  in  full  force.     The  answer 


144 


THE  RATE   OF   GROWTH 


[CH. 


to  this  question  depends  on  whether,  in  the  days  following  the 
first  actual  measurement,  we  can  or  cannot  detect  a  daily  increment 
in  velocity,  before  that  velocity  begins  its  normal  course  of  diminu- 
tion. Now  this  preUminary  ascent  to  a  maximum,  or  point  of 
inflection  of  the  curve,  though  not  shewn  in  the  above-quoted 
experiment,  has  been  often  observed :  as  for  instance,  in  another 
similar  experiment  by  the  author  of  the  former,  the  tadpoles  bein^ 
in  this  case  of  larger  size  (average  49-1  mm.)*. 


Days 

3 

5 

7         10 

12 

14 

17        24 

28        31 

Increment 

0-8f 

)     2-15 

3-66     5^20 

5-95 

6-38 

7-10     7-60 

8-20     8-40 

Or,  by  gra 

,phic  interpolation, 

Days 

Total 
increment 

Daily 
do. 

Days 

Total 
increment 

Daily 
do. 

1 

•23 

•23 

10 

5-29 

•39 

2 

•53 

•30 

11 

5-62 

•33 

3 

■86 

•33 

12 

5^90 

•28 

4 

1-30 

•44 

13 

613 

•23 

5 

200 

•70 

14 

6^38 

•25 

6 

2-78 

•78 

15 

661 

•23 

7 

3-58 

•80 

16 

6-81 

•20 

8 

4-30 

•72 

17 

7-00 

•19  etc. 

9 

4-90 

•60 

. 

The  acceleration  curve  is  drawn  in  Fig.  39. 

Here  we  have  just  what  we  lacked  in  the  former  case,  namely 
a  visible  point  of  inflection  in  the  curve  about  the  seventh  day 
(Figs.  38,  39),  whose  existeiice  is  confirmed  by  successive  observa- 
tions on  the  3rd,  5th,  7th  and  10th  days,  and  which  justifies  to 
some  extent  our  extrapolation  for  the  otherwise  unknown  period 
up  to  and  ending  with  the  third  day;  but  even  here  there  is  a 
short  space  near  the  very  beginning  during  which  we  are  not 
quite  sure  of  the  precise  slope  of  the  curve. 

We  have  now  learned  that,  according  to  these  experiments, 
with  which  many  others  are  in  substantial  agreement,  the  rate  of 
growth  in  the  regenerative  process  is  as  follows.  After  a  very 
short  latent  period,  not  yet  actually  proved  but  whose  existence 
is  highly  probable,  growth  commences  with  a  velocity  which  very 

*  Op.  cit.  p.  406,  Exp.  IV. 


ml     REGENERATION,   OR   C4R0WTH   AND   REPAIR    145 

rapidly  increases  to  a  maximum.  The  curve  quickly, — almost 
suddenly, — changes  its  direction,  as  the  velocity  begins  to  fall; 
and  the  rate  of  fall,  that  is,  the  negative  acceleration,  proceeds 
at  a  slower  and  slower  rate,  which  rate  varies  inversely  as  some 
power  of  the  time,  and  is  found  in  both  of  the  above-quoted 
experiments  to  be  very  approximately  as  1/T'^.  But  it  is  obvious 
that  the  value  which  we  have  found  for  the  latter  portion  of  the 
curve  (however  closely  it  be  conformed  to)  is  only  an  empirical 
value ;   it  has  only  a  temporary  usefulness,  and  must  in  time  give 


0         2        4         6         8 
Fig.  39.     Daily  increment,  or  amount  regenerated,  corresponding  to  Fig.  38. 


place  to  a  formula  which  shall  represent  the  entire  phenomenon, 
from  start  to  finish. 

While  the  curve  of  regenerative  growth  is  apparently  different 
from  the  curve  of  ordinary  growth  as  usually  drawn  (and  while 
this  apparent  difference  has  been  commented  on  and  treated  as 
valid  by  certain  writers)  we  are  now  in  a  position  to  see  that  it 
only  looks  different  because  we  are  able  to  study  it,  if  not  from 
the  beginning,  at  least  very  nearly  so :  while  an  ordinary  curve 
of  growth,  as  it  is  usually  presented  to  us,  is  one  which  dates,  not 

T.  G.  10 


146  THE   RATE    OF   GROWTH  [ch. 

from  the  beginning  of  growth,  but  from  the  comparatively  late, 
and  unimportant,  and  even  fallacious  epoch  of  birth.  A  complete 
curve  of  growth,  starting  from  zero,  has  the  same  essential  charac- 
teristics as  the  regeneration  curve. 

Indeed  the  more  we  consider  the  phenomenon  of  regeneration, 
the  more  plainly  does  it  shew  itself  to  us  as  but  a  particular  case 
of  the  general  phenomenon  of  growth*,  following  the  same  lines, 
obeying  the  same  laws,  and  merely  started  into  activity  by  the 
special  stimulus,  direct  or  indirect,  caused  by  the  infliction  of  a 
wound.  Neither  more  nor  less  than  in  other  problems  of  physiology 
are  we  called  upon,  in  the  case  of  regeneration,  to  indulge  in 
metaphysical  speculation,  or  to  dwell  upon  the  beneficent  purpose 
which  seemingly  underlies  this  process  of  healing  and  restoration. 

It  is  a  very  general  rule,  though  apparently  not  a  universal 
one,  that  regeneration  tends  to  fall  somewhat  short  of  a  complete 
restoration  of  the  lost  part ;  a  certain  percentage  only  of  the  lost 
tissues  is  restored.  This  fact  was  well  known  to  some  of  those 
old  investigators,  who,  like  the  Abbe  Trembley  and  like  Voltaire, 
found  a  fascination  in  the  study  of  artificial  injury  and  the  regenera- 
tion which  followed  it.  Sir  John  Graham  Dalyell,  for  instance, 
says,  in  the  course  of  an  admirable  paragraph  on  regeneration  I : 
"The  reproductive  faculty... is  not  confined  to  one  portion,  but 
may  extend  over  many ;  and  it  may  ensue  even  in  relation  to  the 
regenerated  portion  more  than  once.  Nevertheless,  the  faculty 
gradually  weakens,  so  that  in  general  every  successive  regeneration 
is  smaller  and  more  imperfect  than  the  organisation  preceding  it ; 
and  at  length  it  is  exhausted." 

In  certain  minute  animals,  such  as  the  Infusoria,  in  which  the 
capacity  for  "regeneration"  is  so  great  that  the  entire  animal 
may  be  restored  from  the  merest  fragment,  it  becomes  of  great 
interest  to  discover  whether  there  be  some  definite  size  at  which 
the  fragment  ceases  to  display  this  power.     This  question  has 

*  The  experiments  of  Loeb  on  the  growth  of  Tubulaiia  in  various  saline 
solutions,  referred  to  on  p.  125,  might  as  well  or  better  have  been  referred  to  under 
the  headmg  of  regeneration,  as  they  were  performed  on  cut  pieces  of  the  7,oophji;e. 
(Cf.  Morgan,  op.  cii.  p.  35.) 

"f  Powers  of  the  Creator,  i,  p.  7,  1851.  See  also  Rare  and  Remarkable  Animals, 
II,  pp.   17-19,  90,   1847. 


ml      KEGENERATION,   OR   GROWTH  AND   REPAIR    147 

been  studied  by  Lillie*,  who  found  that  in  Stentor,  while  still 
smaller  fragments  were  capable  of  surviving  for  days,  the  smallest 
portions  capable  of  regeneration  were  of  a  size  equal  to  a  sphere  of 
about  80  jj,  in  diameter,  that  is^  to  say  of  a  volume  equal  to  about 
one  twenty-seventh  of  the  average  entire  animal.  He  arrives  at 
the  remarkable  conclusion  that  for  this,  and  for  all  other  species 
of  animals,  there  is  a  "minimal  organisation  mass,"  that  is  to  say 
a  "minimal  mass  of  definite  size  consisting  of  nucleus  and  cyto- 
plasm within  which  the  organisation  of  the  species  can  just  find 
its  latent  expression."  And  in  like  manner,  Boverif  has  shewn 
that  the  fragment  of  a  sea-urchin's  egg  capable  of  growing  up  into 
a  new  embryo,  and  so  discharging  the  complete  functions  of  an 
entire  and  uninjured  ovum,  reaches  its  limit  at  about  one-twentieth 
of  the  original  egg, — other  writers  having  found  a  limit  at  about 
one-fourth.  These  magnitudes,  small  as  they  are,  represent 
objects  easily  visible  under  a  low  power  of  the  microscope,  and  so 
stand  in  a  very  different  category  to  the  minimal  magnitudes  in 
which  life  itself  can  be  manifested,  and  which  we  have  discussed 
in  chapter  II. 

A  number  of  phenomena  connected  with  the  linear  rate  of 
regeneration  are  illustrated  and  epitomised  in  the  accompanying 
diagram  (Fig.  40),  which  I  have  constructed  from  certain  data 
given  by  Ellis  in  a  paper  on  the  relation  of  the  amount  of  tail 
regenerated  to  the  amount  removed,  in  Tadpoles.  These  data  are 
summarised  in  the  next  Table.  The  tadpoles  were  all  very  much 
of  a  size,  about  40  mm. ;  the  average  length  of  tail  was  very  near 
to  26  mm.,  or  65  per  cent,  of  the  whole  body-length;  and  in  four 
series  of  experiments  about  10,  20,  40  and  60  per  cent,  of  the  tail 
were  severally  removed.  The  amount  regenerated  in  successive 
intervals  of  three  days  is  shewn  in  our  table.  By  plotting  the 
actual  amounts  regenerated  against  these  three-day  intervals  of 
time,  we  may  interpolate  values  for  the  time  taken  to  regenerate 
definite  percentage  amounts,  5  per  cent.,  10  per  cent.,  etc.  of  the 

*  Lillie,  F.  R.,  The  smallest  Parts  of  Stentor  capable  of  Regeneration, 
Journ.  of  Morphology,  xii,  p.  23.9,   1897. 

t  Boveri,  Entwicklungsfahigkeit  kernloser  Seeigeleier,  etc.,  Arch.f.  Entw.  Mech. 
II,  1895.  See  also  Morgan,  Studies  of  the  partial  larvae  of  Sphaerechinus,  ibid. 
1895;  J.  Loeb,  On  the  Limits  of  Divisibility  of  Living  Matter,  Biol.  Lectures,  1894, 
etc. 

10—2 


148  THE   RATE   OF   GROWTH 

The  Rate  of  Regenerative  Growth  in  Tadpoles'  Tails. 
{After  M.  M.  Ellis,  J.  Exp.  Zool.  vii,  p.  421,  1909.) 


[CH. 


Body 
length 
mm. 

Tail 
length 
mm. 

Amovmt 

removed 

mm. 

Per  cent. 

of  tail 
removed 

0/ 
,0 

amount  regenerated 

in  days 

Series* 

3 

6 

9      12 

15 

18     32 

0 

39-575 

25-895 

3-2 

12-36 

13 

31 

44     44 

44 

44     44 

P 

40-21 

26-13 

5-28 

20-20 

10 

29 

40    44 

44 

44     44 

R 

39-86 

25-70 

10-4 

40-50 

6 

20 

31     40 

48 

48     48 

8 

40-34 

26-11 

14-8 

56-7 

0 

16 

33     39 

45 

48     48 

*  Each  series  gives  the  mean  of  20  experiments. 


removed 


Fig.  40.  Relation  between  the  percentage  amomit  of  tail  removed,  the  percentage 
restored,  and  the  time  required  for  its  restoration.  (From  M.  M.  Ellis's 
data.) 

amount  removed ;  and  my  diagram  is  constructed  from  the  four 
sets  of  values  thus  obtained,  that  is  to  say  from  the  four  sets  of 
experiments  which  differed  from  one  another  in  the  amount  of 
tail  amputated.  To  these  we  have  to  add  the  general  result  of  a 
fifth  series  of  experiments,  which  shewed  that  when  as  much  as 
75  per  cent,  of  the  tail  was  cut  off,  no  regeneration  took  place  at 
all,  but  the  animal  presently  died.     In  our  diagram,  then,  each 


Ill]    REGENERATION,   OR   GROWTH  AND   REPAIR     149 

curve  indicates  the  time  taken  to  regenerate  n  per  cent,  of  the 
amount  removed.  All  the  curves  converge  towards  infinity,  when 
the  amount  removed  (as  shewn  by  the  ordinate)  approaches  75 
per  cent. ;  and  all  of  the  curves  start  from  zero,  for  nothing  is 
regenerated  where  nothing  had  been  removed.  Each  curve  ap- 
proximates in  form  to  a  cubic  parabola. 

The  amount  regenerated  varies  also  with  the  age  of  the  tadpole 
and  with  other  factors,  such  as  temperature;  in  other  words,  for 
any  given  age,  or  size,  of  tadpole  and  also  for  various, specific 
temperatures,  a  similar  diagram  might  be  constructed. 

The  power  of  reproducing,  or  regenerating,  a  lost  hmb  is 
particularly  well  developed  in  arthropod  animals,  and  is  some- 
times accompanied  by  remarkable  modification  of  the  form  of 
the  regenerated  limb.  A  case  in  point,  which  has  attracted 
much  attention,  occurs  in  connection  with  the  claws  of  certain 
Crustacea*. 

In  many  Crustacea  we  have  an  asymmetry  of  the  great  claws, 
one  being  larger  than  the  other  and  also  more  or  less  different  in 
form.  For  instance,  in  the  common  lobster,  one  claw,  the  larger 
of  the  two,  is  provided  with  a  few  great  ''crushing"  teeth,  while 
the  smaller  claw  has  more  numerous  teeth,  small  and  serrated. 
Though  Aristotle  thought  otherwise,  it  appears  that  the  crushing- 
claw  may  be  on  the  right  or  left  side,  indifferently ;  whether  it 
be  on  one  or  the  other  is  a  problem  of  "chance."  It  is  otherwise 
in  many  other  Crustacea,  where  the  larger  and  more  powerful 
claw  is  always  left  or  right,  as  the  case  may  be,  according  to  the 
species:  where,  in  other  words,  the  "probability"  of  the  large 
or  the  small  claw  being  left  or  being  right  is  tantamount  to 
certainty}". 

The  one  claw  is  the  larger  because  it  has  grown  the  faster; 

*  Cf.  Przibram,  H.,  Scheerenumkehr  bei  dekapoden  Crustaceen,  Arch.  f.  Entw. 
Mech.  XIX,  181-247,  1905;  xxv,  266-344,  1907.  Emmel,  ibid,  xxii,  542,  1906; 
Regeneration  of  lost  parts  in  Lobster,  Rep.  Comm.  Inland  Fisheries,  Rhode  Island, 
XXXV,  xxxvi,  1905-6;  Science  (n.s.),  xxvi,  83-87,  1907.  Zeleny,  Compensatory 
Regulation,  J.  Exp.  Zool.  n,  1-102,  347-369,  1905;   etc. 

t  Lobsters  are  occasionally  found  with  two  symmetrical  claws :  which  are  then 
usually  serrated,  sometimes  (but  very  rai-ely)  both  blunt-toothed.  Cf.  Caiman, 
P.Z.S.  1906,  pp.  633,  634,  and  reff. 


150  THE   RATE   OF   CxROWTH  [ch. 

it  has  a  higher  "coefficieDt  of  growth,"  and  accordingly,  as  age 
advances,  the  disproportion  between  the  two  claws  becomes  more 
and  more  evident.  Moreover,  we  must  assume  that  the  character- 
istic form  of  the  claw  is  a  "function"  of  its  magnitude;  the 
knobbiness  is  a  phenomenon  coincident  with  growth,  and  we 
never,  under  any  circumstances,  find  the  smaller  claw  with  big 
crushing  teeth  and  the  big  claw  with  little  serrated  ones.  There 
are  many  other  somewhat  similar  cases  where  size  and  form  are 
manifestly  correlated,  and  we  have  already  seen,  to  some  extent, 
that  th^  phenomenon  of  growth  is  accompanied  by  certain  ratios 
of  velocity  that  lead  inevitably  to  changes  of  form.  Meanwhile, 
then,  we  must  simply  assume  that  the  essential  difference  between 
the  two  claws  is  one  of  magnitudej  with  which  a  certain  difEerentia- 
tion  of  form  is  inseparably  associated. 

If  we  amputate  a  claw,  or  if,  as  often  happens,  the  crab  "casts 
it  off,"  it  undergoes  a  process  of  regeneration, — it  grows  anew, 
and  evidently  does  so  with  an  accelerated  velocity,  which  accelera- 
tion will  cease  when  equilibrium  of  the  parts  is  once  more  attained : 
the  accelerated  velocity  being  a  case  in  point  to  illustrate  that 
vis  revulsionis  of  Haller,  to  which  we  have  already  referred. 

With  the  help  of  this  principle,  Przibram  accounts  for  certain 
curious  phenomena  which  accompany  the  process  of  regeneration. 
As  his  experiments  and  those  of  Morgan  shew,  if  the  large  or 
knobby  claw  (A)  be  removed,  there  are  certain  cases,  e.g.  the 
common  lobster,  where  it  is  directly  regenerated.  In  other  cases, 
e.g.  Alpheus*,  the  other  claw  (B)  assumes  the  size  and  form  of  that 
which  was  amputated,  while  the  latter  regenerates  itself  in  the 
form  of  the  other  and  weaker  one;  A  and  B  have  apparently 
changed  places.  In  a  third  case,  as  in  the  crabs,  the  yl-claw  re- 
generates itself  as  a  small  or  5-claw,  but  the  B-claw  remains  for  a 
time  unaltered,  though  slowly  and  in  the  course  of  repeated  moults 
it  later  on  assumes  the  large  and  heavily  toothed  ^-form. 

Much  has  been  written  on  this  phenomenon,  but  in  essence  it 
is  very  simple.  It  depends  upon  the  respective  rates  of  growth, 
upon  a  ratio  between  the  rate  of  regeneration  and  the  rate  of 
growth  of  the  uninjured  limb :    complicated  a  little,  however,  by 

*  Wilson,  E.  B.,  Reversal  of  Symmetry  in  Alpheus  heterochelis,  Biol.  Bull,  rv, 
p.  197,  1903. 


Ill]    REGENERATION,   OR   GROWTH   AND   REPAIR    151 

the  possibility  of  the  uninjured  Hmb  growing  all  the  faster  for 
a  time  after  the  animal  has  been  relieved  of  the  other.  From  the 
time  of  amputation,  say  of  ^,  ^  begins  to  grow  from  zero,  with 
a  high  "regenerative"  velocity;  while  B,  starting  from  a  definite 
magnitude,  continues  to  increase,  with  its  normal  or  perhaps 
somewhat  accelerated  velocity.  The  ratio  between  the  two 
velocities  of  growth  will  determine  whether,  by  a  given  time, 
A  has  equalled,  outstripped,  or  still  fallen  short  of  the  magnitude 
of  i5. 

That  this  is  the  gist  of  the  whole  problem  is  confirmed  (if 
confirmation  be  necessary)  by  certain  experiments  of  Wilson's. 
It  is  known  that  by  section  of  the  nerve  to  a  crab's  claw,  its 
growth  is  retarded,  and  as  the  general  growth  of  the  animal 
proceeds  the  claw  comes  to  appear  stunted  or  dwarfed.  Now  in 
such  a  case  as  that  of  Alpheus,  we  have  seen  that  the  rate  of 
regenerative  growth  in  an  amputated  large  claw  fails  to  let  it 
reach  or  overtake  the  magnitude  of  the  growing  little  claw : 
which  latter,  in  short,  now  appears  as  the  big  one.  But  if  at  the 
same  time  as  we  amputate  the  big  claw  we  also  sever  the  nerve 
to  the  lesser  one,  we  so  far  slow  down  the  latter's  growth  that 
the  other  is  able  to  make  up  to  it,  and  in  this  case  the  two  claws 
continue  to  grow  at  approximately  equal  rates,  or  in  other  words 
continue  of  coequal  size. 

The  phenomenon  of  regeneration  goes  some  way  towards 
helping  us  to  comprehend  the  phenomenon  of  "multiplication  by 
fission,"  as  it  is  exemplified  at  least  in  its  simpler  cases  in  many 
worms  and  worm-like  animals.  For  physical  reasons  which  we 
shall  have  to  study  in  another  chapter,  there  is  a  natural  tendency 
for  any  tube,  if  it  have  the  properties  of  a  fluid  or  semi-fluid 
substance,  to  break  up  into  segments  after  it  comes  to  a  certain 
length ;  and  nothing  can  prevent  its  doing  so,  except  the  presence 
of  some  controlling  force,  such  for  instance  as  may  be  due  to  the 
pressure  of  some  external  support,  or  some  superficial  thickening 
or  other  intrinsic  rigidity  of  its  own  substance.  If  we  add  to  this 
natural  tendency  towards  fission  of  a  cylindrical  or  tubular  worm, 
the  ordinary  phenomenon  of  regeneration,  we  have  all  that  is 
essentially  implied  in  "reproduction  by  fission."     And  in  so  far 


152  THE  RATE   OF   GROWTH  [ch. 

as  the  process  rests  upon  a  physical  principle,  or  natural  tendency, 
we  may  account  for  its  occurrence  in  a  great  variety  of  animals, 
zoologically  dissimilar ;  and  also  for  its  presence  here  and  absence 
there,  in  forms  which,  though  materially  different  in  a  physical 
sense,  are  zoologically  speaking  very  closely  allied. 

Conclusion  and  Summary. 

But  the  phenomena  of  regeneration,  like  all  the  other 
phenomena  of  growth,  soon  carry  us  far  afield,  and  we  must  draw 
this  brief  discussion  to  a  close. 

For  the  main  features  which  appear  to  be  common  to  all 
curves  of  growth  we  may  hope  to  have,  some  day,  a  physical 
explanation.  In  particular  we  should  like  to  know  the  meaning 
of  that  point  of  inflection,  or  abrupt  change  from  an  increasing 
to  a  decreasing  velocity  of  growth  which  all  our  curves,  and 
especially  our  acceleration  curves,  demonstrate  the  existence  of, 
provided  only  that  they  include  the  initial  stages  of  the  whole 
phenomenon:  just  as  we  should  also  like  to  have  a  full  physical 
or  physiological  explanation  of  the  gradually  diminishing  velocity 
of  growth  which  follows,  and  which  (though  subject  to  temporary 
interruption  or  abeyance)  is  on  the  whole  characteristic  of  growth  in 
all  cases  whatsoever.  In  short,  the  characteristic  form  of  the  curve 
of  growth  in  length  (or  any  other  linear  dimension)  is  a  phenomenon 
which  we  are  at  present  unable  to  explain,  but  which  presents 
us  with  a  definite  and  attractive  problem  for  future  solution. 
It  would  seem  evident  that  the  abrupt  change  in  velocity  must  be 
due,  either  to  a  change  in  that  pressure  outwards  from  within, 
by  which  the  "forces  of  growth"  make  themselves  manifest,  or 
to  a  change  in  the  resistances  against  which  they  act,  that  is  to 
say  the  tension  of  the  surface ;  and  this  latter  force  we  do  not  by 
any  means  limit  to  "surface-tension"  proper,  but  may  extend  to 
the  development  of  a  more  or  less  resistant  membrane  or  "skin," 
or  even  to  the  resistance  of  fibres  or  other  histological  elements, 
binding  the  boundary  layers  to  the  parts  within.  I  take  it  that 
the  sudden  arrest  of  velocity  is  much  more  hkely  to  be  due  to  a 
sudden  increase  of  resistance  than  to  a  sudden  diminution  of 
internal  energies :  in  other  words,  I  suspect  that  it  is  coincident 
with  some  notable  event  of  histological  differentiation,  such  as 


Ill]  CONCLUSION  AND   SUMMARY  153 

the  rapid  formation  of  a  comparatively  firm  skin;  and  that  the 
dwindUng  of  velocities,  or  the  negative  acceleration,  which  follows, 
is  the  resultant  or  composite  effect  of  waning  forces  of  growth  on 
the  one  hand,  and  increasing  superficial  resistance  on  the  other. 
This  is  as  much  as  to  say  that  growth,  while  its  own  energy  tends 
to  increase,  leads  also,  after  a  while,  to  the  establishment  of 
resistances  which  check  its  own  further  increase. 

Our  knowledge  of  the  whole  complex  phenomenon  of  growth 
is  so  scanty  that  it  may  seem  rash  to  advance  even  this  tentative 
suggestion.  But  yet  there  are  one  or  two  known  facts  which 
seem  to  bear  upon  the  question,  and  to  indicate  at  least  the  manner 
in  which  a  varying  resistance  to  expansion  may  afiect  the  velocity 
of  growth.  For  instance,  it  has  been  shewn  by  Frazee*  that 
electrical  stimulation  of  tadpoles,  with  small  current  density  and 
low  voltage,  increases  the  rate  of  regenerative  growth.  As  just 
such  an  electrification  would  tend  to  lower  the  surface-tension, 
and  accordingly  decrease  the  external  resistance,  the  experiment 
would  seem  to  support,  in  some  slight  degree,  the  suggestion 
which  I  have  made. 

Delagef  has  lately  made  use  of  the  principle  of  specific  rate  of  growth, 
in  considering  the  question  of  heredity  itself.  We  know  that  the  chromatin 
of  the  fertilised  egg  comes  from  the  male  and  female  parent  alike,  in  equal  or 
nearly  equal  shares;  we  know  that  the  initial  chromatin,  so  contributed, 
multiplies  many  thousand-fold,  to  supply  the  chromatin  for  every  cell  of  the 
offspring's  body;  and  it  has,  therefore,  a  high  "coefficient  of  growth."  If  we 
admit,  with  Van  Beneden  and  others,  that  the  initial  contributions  of  male  and 
female  chromatin  continue  to  be  transmitted  to  the  succeeding  generations 
of  cells,  we  may  then  conceive  these  chromatins  to  retain  each  its  own  coefficient 
of  growth ;  and  if  these  differed  ever  so  little,  a  gradual  preponderance  of  one 
or  other  would  make  itself  felt  in  time,  and  might  conceivably  explain  the 
preponderating  influence  of  one  parent  or  the  other  upon  the  characters  of 
the  offspring.  Indeed  O.  Hertwig  is  said  (according  to  Delage's  interpretation) 
to  have  actually  shewn  that  we  can  artificially  modify  the  rate  of  growth  of 
one  or  other  chromatin,  and  so  increase  or  diminish  the  influence  of  the  maternal 
or  paternal  heredity.  This  theory  of  Delage's  has  its  fascination,  but  it  calls 
for  somewhat  large  assumptions;  and  in  particular,  it  seems  (Kke  so  many 
other  theories  relating  to  the  chromosomes)  to  rest  far  too  much  upon  material 
elements,  rather  than  on  the  imponderable  dynamic  factors  of  the  cell. 

*  J.  Exp.  Zool.  VII,  p.  457,  1909. 
f  Biologica,  ni,  p.  161,  June,.  1913. 


154  THE  KATE   OF   GROWTH  [ch. 

We  may  summarise,  as  follows,  the  main  results  of  the  fore- 
going discussion : 

(1)  Except  in  certain  minute  organisms  and  minute  parts  of 
organisms,  whose  form  is  due  to  the  direct  action  of  molecular 
forces,  we  may  look  upon  the  form  of  the  organism  as  a  "function 
of  growth,"  or  a  direct  expression  of  a  rate  of  growth  which  varies 
according  to  its  different  directions. 

(2)  Rate  of  growth  is  subject  to  definite  laws,  and  the 
velocities  in  different  directions  tend  to  maintain  a  ratio  which  is 
more  or  less  constant  for  each  specific  organism ;  and  to  this 
regularity  is  due  the  fact  that  the  form  of  the  organism  is  in  general 
regular  and  constant. 

(3)  Nevertheless,  the  ratio  of  velocities  in  different  directions 
is  not  absolutely  constant,  but  tends  to  alter  or  fluctuate  in  a 
regular  way ;  and  to  these  progressive  changes  are  due  the 
changes  of  form  which  accompany  "development,"  and  the  slower 
changes  of  form  which  continue  perceptibly  in  after  life. 

(4)  The  rate  of  growth  is  a  function  of  the  age  of  the  organism  , 
it  has  a  maximum  somewhat  early  in  life,  after  which  epoch  of 
maximum  it  slowly  declines. 

(5)  The  rate  of  growth  is  directly  affected  by  temperature, 
and  by  other  physical  conditions. 

(6)  It  is  markedly  affected,  in  the  way  of  acceleration  or 
retardation,  at  certain  physiological  epochs  of  life,  such  as  birth, 
puberty,  or  metamorphosis. 

(7)  Under  certain  circumstances,  growth  may  be  negative,  the 
organism  growing  smaller :  and  such  negative  growth  is  a  common 
accompaniment  of  metamorphosis,  and  a  frequent  accompaniment 
of  old  age. 

(8)  The  phenomenon  of  regeneration  is  associated  with  a  large 
temporary  increase  in  the  rate  of  growth  (or  "acceleration''  of 
growth)  of  the  injured  surface ;  in  other  respects,  regenerative 
growth  is  similar  to  ordinary  growth  in  all  its  essential  phenomena. 

In  this  discussion  of  grbwth,  we  have  left  out  of  account  a 
vast  number  of  processes,  or  phenomena,  by  which,  in  the  physio- 
logical mechanism  of  the  body,  growth  is  effected  and  controlled. 
We  have  dealt  with  growth  in  its  relation  to  magnitude,  and  to 


ml  CONCLUSION  AND   SUMMARY  155 

that  relativity  of  magnitudes  which  constitutes  form ;  and  so  we 
have  studied  it  as  a  phenomenon  which  stands  at  the  beginning 
of  a  morphological,  rather  than  at  the  end  of  a  physiological 
enquiry.  Under  these  restrictions,  we  have  treated  it  as  far  as 
possible,  or  in  such  fashion  as  our  present  knowledge  permits,  on 
strictly  physical  lines. 

In  all  its  aspects,  and  not  least  in  its  relation  to  form,  the 
growth  of  organisms  has  many  analogies,  some  close  and  some 
perhaps  more  remote,  among  inanimate  things.  As  the  waves 
grow  when  the  winds  strive  with  the  other  forces  which  govern 
the  movements  of  the  surface  of  the  sea,  as  the  heap  grows  when 
we  pour  corn  out  of  a  sack,  as  the  crystal  grows  when  from  the 
surrounding  solution  the  proper  molecules  fall  into  their  appro- 
priate places :  so  in  all  these  cases,  very  much  as  in  the  organism 
itself,  is  growth  accompanied  by  change  of  form,  and  by  a  develop- 
ment of  definite  shapes  and  contours.  And  in  these  cases  (as 
in  all  other  mechanical  phenomena),  we  are  led  to  equate  our 
various  magnitudes  with  time,  and  so  to  recognise  that  growth  is 
essentially  a  question  of  rate,  or  of  velocity. 

The  differences  of  form,  and  changes  of  form,  which  are  brought 
about  by  varying  rates  (or  "laws")  of  growth,  are  essentially  the 
same  phenomenon  whether  they  be,  so  to  speak,  episodes  in  the 
life-history  of  the  individual,  or  manifest  themselves  as  the  normal 
and  distinctive  characteristics  of  what  we  call  separate  species  of 
the  race.  From  one  form,  or  ratio  of  magnitude,  to  another  there 
is  but  one  straight  and  direct  road  of  transformation,  be  the 
journey  taken  fast  or  slow;  and  if  the  transformation  take  place 
at  all,  it  will  in  all  hkelihood  proceed  in  the  self -same  way,  whether 
it  occur  within  the  life-time  of  an  individual  or  during  the  long 
ancestral  history  of  a  race.  No  small  part  of  what  is  known  as 
Wolff's  or  von  Baer's  law,  that  the  individual  organism  tends  to 
pass  through  the  phases  characteristic  of  its  ancestors,  or  that  the 
life-history  of  the  individual  tends  to  recapitulate  the  ancestral 
history  of  its  race,  lies  wrapped  up  in  this  simple  account  of  the 
relation  between  rate  of  growth  and  form. 

But  enough  of  this  discussion.  Let  us  leave  for  a  while  the 
subject  of  the  growth  of  the  organism,  and  attempt  to  study  the 
conformation,  within  and  without,  of  the  individual  cell. 


CHAPTER  IV 

ON  THE  INTERNAL  FORM  AND  STRUCTURE  OF  THE  CELL 

In  the  early  days  of  the  cell-theory,  more  than  seventy  years 
ago,  Goodsir  was  wont  to  speak  of  cells  as  "centres  of  growth" 
or  "centres  of  nutrition,"  and  to  consider  them  as  essentially 
"centres  of  force."  He  looked  forward  to  a  time  when  the  forces 
connected  with  the  cell  should  be  particularly  investigated :  when, 
that  is  to  say,  minute  anatomy  should  be  studied  in  its  dynamical 
aspect.  "When  this  branch  of  enquiry,"  he  says  "shall  have 
been  opened  up,  we  shall  expect  to  have  a  science  of  organic 
forces,  having  direct  relation  to  anatomy,  the  science  of  organic 
•forms*."  And  likewise,  long  afterwards,  Giard  contemplated  a 
science  of  mor'phodynamique, — but  still  looked  upon  it  as  forming 
so  guarded  and  hidden  a  "territoire  scientifique,  que  la  plupart 
des  naturalistes  de  nos  jours  ne  le  verront  que  comme  Moise  vit 
la  terre  promise,  seulement  de  loin  et  sans  pouvoir  y  entrerf." 

To  the  external  forms  of  cells,  and  to  the  forces  which  produce 
and  modify  these  forms,  we  shall  pay  attention  in  a  later  chapter. 
But  there  are  forms  and  configurations  of  matter  within  the  cell, 
which  also  deserve  to  be  studied  with  due  regard  to  the  forces, 
known  or  unknown,  of  whose  resultant  they  are  the  visible 
expression. 

In  the  long  interval  since  Goodsir's  day,  the  visible  structure, 
the  conformation  and  configuration,  of  the  cell,  has  been  studied 
far  more  abundantly  than  the  purely  dynamic  problems  that  are 
associated  therewith.  The  overwhelming  progress  of  microscopic 
observation  has  multiplied  our  knowledge  of  cellular  and  intra- 
cellular structure ;    and  to  the  multitude  of  visible  structures  it 

*  Anatomical  and  Pathological  Observations,  p.  3,  1845;  Anatomical  Memoirs, 
n,  p.  392,  18G8. 

t  Giard.  A.,  L'oeuf  et  les  debuts  de  revolution,  Bull.  Sci.  du  Nord  de  la  Fr. 
VIII,  pp.  252-258,  1876. 


CH.  IV]    INTERNAL  FORM  AND  STRUCTURE  OF  CELL    L57 

has  been  often  easier  to  attribute  virtues  than  to  ascribe  intelligible 
functions  or  modes  of  action.  But  here  and  there  nevertheless, 
throughout  the  whole  literature  of  the  subject,  we  find  recognition 
of  the  inevitable  fact  that  dynamical  problems  lie  behind  the 
morphological  problems  of  the  cell. 

Blitschli  pointed  out  forty  years  ago,  with  emphatic  clearness, 
the  failure  of  morphological  methods,  and  the  need  for  physical 
methods,  if  we  were  to  penetrate  deeper  into  the  essential  nature 
of  the  cell*.  And  such  men  as  Loeb  and  Whitman,  Driesch  and 
Roux,  and  not  a  few  besides,  have  pursued  the  same  train  of 
thought  and  similar  methods  of  enquiry. 

Whitman!,  for  instance,  puts  the  case  in  a  nutshell  when,  in 
speaking  of  the  so-called  "  caryokinetic "  phenomena  of  nuclear 
division,  he  reminds  us  that  the  leading  idea  in  the  term  "  caryo- 
kinesis''  is  motion, — "motion  viewed  as  an  exponent  of  forces 
residing  in,  or  acting  upon,  the  nucleus.  It  regards  the  nucleus 
as  a  seat  of  energy,  which  displays  itself  in  -phenomena  of  motion  J." 

In  short  it  would  seem  evident  that,  except  in  relation  to  a 
dynamical  investigation,  the  mere  study  of  cell  structure  has  but 
little  value  of  its  own.  That  a  given  cell,  an  ovum  for  instance, 
contains  this  or  that  visible  substance  or  structure,  germinal 
vesicle  or  germinal  spot,  chromatin  or  achromatin,  chromosomes 
or  centrosomes,  obviously  gives  no  explanation  of  the  activities  of 
the  cell.  And  in  all  such  hypotheses  as  that  of  "pangenesis,"  in 
all  the  theories  which  attribute   specific  properties  to  micellae, 

*  Entwickelungsvorgdnge  der  Eizelle,  1876;  Investigations  on  Microscopic  Foams 
and  Protoplasm,  p.  1,  1894. 

t  Journ.  of  Morphology,  i,  p.  229,  1887. 

J  While  it  has  been  very  common  to  look  upon  the  phenomena  of  mitosis  as- 
sufficiently  explained  by  the  results  towards  which  they  seem  to  lead,  we  may  iind 
here  and  there  a  strong  protest  against  this  mode  of  interpretation.  The  following 
is  a  case  in  point:  "On  a  tente  d'etablir  dans  la  mitose  dite  primitive  plusieurs 
categories,  plusieurs  types  de  mitose.  On  a  choisi  le  plus  souvent  comme  base 
de  ces  systemes  des  concepts  abstraits  et  teleologiques :  repartition  plus  ou  moins 
exacte  de  la  chromatine  entre  les  deux  noyaux-fils  suivant  qu'il  y  a  ou  non  des 
chromosomes  (Da.ngeard),  distribution  particuliere  et  signification  dualiste  des 
substances  nucleaires  (substance  kmetique  et  substance  generative  ou  hereditaire, 
Hartmann  et  ses  eleves),  etc.  Pour  moi  tous  ces  essais  sont  a  rejeter  categorique- 
ment  a  cause  de  leur  caractere  finaliste ;  de  plus,  ils  sont  construits  sur  des  concepts 
non  demontres,  et  qui  parfois  representent  des  generalisations  absolument  erronees." 
A.  AlexeiefE,  Archivfitr  Protistenkunde,  xix,  p.  344,  1913. 


158  ON   THE   INTERNAL   FORM  AND  [ch. 

idioplasts,  ids,  or  other  constituent  particles  of  protoplasm  or  of 
the  cell,  we  are  apt  to  fall  into  the  error  of  attributing  to  matter 
what  is  due  to  energy  and  is  manifested  in  force  :  or,  more  strictly 
speaking,  of  attributing  to  material  particles  individually  what  is 
due  to  the  energy  of  their  collocation. 

The  tendency  is  a  very  natural  one,  as  knowledge  of  structure 
increases,  to  ascribe  particular  virtues  to  the  material  structures 
themselves,  and  the  error  is  one  into  which  the  disciple  is  likely 
to  fall,  but  of  which  we  need  not  suspect  the  master-mind.  The 
dynamical  aspect  of  the  case  was  in  all  probability  kept  well  in 
view  by  those  who,  like  Goodsir  himself,  first  attacked  the  problem 
of  the  cell  and  originated  our  conceptions  of  its  nature  and 
functions. 

But  if  we  speak,  as  Weismann  and  others  speak,  of  an 
"hereditary  substance/''  a  substance  which  is  split  off  from  the 
parent-body,  and  which  hands  on  to  the  new  generation  the 
characteristics  of  the  old,  we  can  only  justify  our  mode  of  speech 
by  the  assumption  that  that  particular  portion  of  matter  is  the 
essential  vehicle  of  a  particular  charge  or  distribution  of  energy, 
in  which  is  involved  the  capability  of  producing  motion,  or  of 
doing  "work." 

For,  as  Newton  said,  to  tell  us  that  a  thing  "is  endowed  with 
an  occult  specific  quality,  by  which  it  acts  and  produces  manifest 
effects,  is  to  tell  us  nothing;  but  to  derive  two  or  three  general 
principles  of  motion*  from  phenomena  would  be  a  very  great  step 
in  philosophy,  though  the  causes  of  these  principles  were  not  yet 
discovered."  The  things  which  we  see  in  the  cell  are  less  important 
than  the  actions  which  we  recognise  in  the  cell;  and  these  latter 
we  must  especially  scrutinize,  in  the  hope  of  discovering  how  far 
they  may  be  attributed  to  the  simple  and  well-known  physical 
forces,  and  how  far  they  be  relevant  or  irrelevant  to  the  phenomena 
which  we  associate  with,  and  deem  essential  to,  the  manifestation 
of  life.  It  may  be  that  in  this  way  we  shall  in  time  draw  nigh  to 
the  recognition  of  a  specific  and  ultimate  residuum. 

*  This  is  the  old  philosophic  axiom  writ  large:  Ignorato  motu,  ignoratur 
natura ;  which  again  is  but  an  adaptation  of  Aristotle's  phrase,  17  dpxv  rrjs  Kivr]<T€us. 
as  equivalent  to  the  "Efficient  Cause."  FitzGerald  holds  that  ''all  explanation 
consists  in  a  desciiption  of  underlying  motions";  Scientific  Writings,  1902,  p.  385. 


IV]  STRUCTURE   OF   THE   CELL  159 

And  lacking,  as  we  still  do  lack,  direct  knowledge  of  the  actual 
forces  inherent  in  the  cell,  we  may  yet  learn  something  of  their 
distribution,  if  not  also  of  their  nature,  from  the  outward  and 
inward  configuration  of  the  cell,  and  from  the  changes  taking 
place  in  this  configuration ;  that  is  to  say  from  the  movements 
of  matter,  the  kinetic  phenomena,  which  the  forces  in  action  set  up. 

The  fact  that  the  germ-cell  develops  into  a  very  complex 
structure,  is  no  absolute  proof  that  the  cell  itself  is  structurally 
a  very  comphcated  mechanism :  nor  yet,  though  this  is  somewhat 
less  obvious,  is  it  sufficient  to  prove  that  the  forces  at  work,  or 
latent,  within  it  are  especially  numerous  and  complex.  If  we  blow 
into  a  bowl  of  soapsuds  and  raise  a  great  mass  of  many-hued  and 
variously  shaped  bubbles,  if  we  explode  a  rocket  and  watch  the 
regular  and  beautifiil  configuration  of  its  falUng  streamers,  if  we 
consider  the  wonders  of  a  limestone  cavern  which  a  filtering  stream 
has  filled  with  stalactites,  we  soon  perceive  that  in  all  these  cases 
we  have  begun  with  an  initial  system  of  very  slight  complexity, 
whose  structure  in  no  way  foreshadowed  the  result,  and  whose 
comparatively  simple  intrinsic  forces  only  play  their  part  by 
complex  interaction  with  the  equally  simple  forces  of  the  surround- 
ing medium.  In  an  earlier  age,  men  sought  for  the  visible  embryo, 
even  for  the  homunculus,  within  the  reproductive  cells ;  and  to 
this  day,  we  scrutinize  these  cells  for  visible  structure,  unable  to 
free  ourselves  from  that  old  doctrine  of  "pre-formation*." 

Moreover,  the  microscope  seemed  to  substantiate  the  idea 
(which  we  may  trace  back  to  Leibniz |  and  to  HobbesJ),  that 
there  is  no  limit  to  the  mechanical  complexity  which  we  may 
postulate  in  an  organism,  and  no  limit,  therefore,  to  the  hypo- 
theses which  we  may  rest  thereon. 

But  no  microscopical  examination  of  a  stick  of  sealing-wax, 
no  study  of  the  material  of  which  it  is  composed,  can  enlighten 

*  As  when  Nageli  concluded  that  the  organism  is,  in  a  certain  sense,  "vorge- 
bildet" ;  Beitr.  zur  iviss.  Botanik,  II,  1860.    Cf.  E.  B.  Wilson,  The  Cell,  etc.,  p.  302. 

t  "La  matiere  arrangee  par  une  sagesse  divine  doit  etre  essentieUement  organisee 
partout...il  y  a  machine  dans  les  parties  de  la  machine  Naturelle  a  I'infini."  Sur  le 
qirincipe  de  la  Vie,  p.  431  (Erdmann).  This  is  the  very  converse  of  the  doctrine 
of  the  Atomists,  who  could  not  conceive  a  condition  "w6i  dimidiae  jjartis  pars 
■semper  habebit  Dimidiam  partem,  nee  res  praefiniet  ulla.''' 

1  Cf.  an  interesting  passage  from  the  Elements  (i,  p.  445,  Molesworth's  edit.), 
quoted  by  Owen,  Hunterian  Lectures  on  the  Invertebrates,  2nd  ed.  pp.  40,  41,  1855. 


160  ON   THE  INTERNAL  FORM  AND  [ch. 

us  as  to  its  electrical  manifestations  or  properties.  Matter  of 
itself  has  no  power  to  do,  to  make,  or  to  become :  it  is  in  energy 
that  all  these  potentialities  reside,  energy  invisibly  associated  with 
the  material  system,  and  in  interaction  with  the  energies  of  the 
surrounding  universe. 

That  "function  presupposes  structure"  has  been  declared  an 
accepted  axiom  of  biology.  Who  it  was  that  so  formulated  the 
aphorism  I  do  not  know ;  but  as  regards  the  structure  of  the  cell 
it  harks  back  to  Briicke,  with  whose  demand  for  a  mechanism, 
or  organisation,  within  the  cell  histologists  have  ever  since 
been  attempting  to  comply*.  But  unless  we  mean  to  include 
thereby  invisible,  and  merely  chemical  or  molecular,  structure, 
we  come  at  once  on  dangerous  ground.  For  we  have  seen,  in 
a  former  chapter,  that  some  minute  "organisms"  are  already 
known  of  such  all  but  infinitesimal  magnitudes  that  everything 
which  the  morphologist  is  accustomed  to  conceive  as  "structure" 
has  become  physically  impossible ;  and  moreover  recent  research 
tends  generally  to  reduce,  rather  than  to  extend,  our  conceptions 
of  the  visible  structure  necessarily  inherent  in  living  protoplasm. 
The  microscopic  structure  which,  in  the  last  resort  or  in  the  simplest 
cases,  it  seems  to  shew,  is  that  of  a  more  or  less  viscous  colloid, 
or  rather  mixture  of  colloids,  and  nothing  more.  Now,  as  Clerk 
Maxwell  puts  it,  in  discussing  this  very  problem,  "one  material 
system  can  differ  from  another  only  in  the  configuration  and 
motion  which  it  has  at  a  given  instant  t-"  If  we  cannot  assume 
differences  in  structure,  we  must  assume  differences  in  niotion,  that 
is  to  say,  in  energy.  And  if  we  cannot  do  this,  then  indeed  we  are 
thrown  back  upon  modes  of  reasoning  unauthorised  in  physical 
science,  and  shall  find  ourselves  constrained  to  assume,  or  to 
"admit,  that  the  properties  of  a  germ  are  not  those  of  a  purely 
material  system." 

*  "Wir  miissen  deshalb  den  lebenden  Zellen,  abgesehen  von  der  Molekular- 
structur  der  organischen  Verbindungen  welche  sie  enthalt,  noch  eine  andere  und 
in  anderer  Weise  complicirte  Structur  zuschreiben,  und  diese  es  ist  welche  wir 
mit  dem  Namen  Organisation  bezeiclmen,"  Briicke^  Die  Elementarorganismen, 
Wiener  Sitzungsber.  xliv,  1861,  p.  386;  quoted  by  Wilson,  The  Cell,  etc.  p.  289. 
Cf.  also  Hardy,  Journ.  of  Physiol,  xxiv,  1899,  p.  159. 

f  Precisely  as  in  the  Lucretian  concursus,  motus,  ordo,  positura,figurae,  whereby 
bodies  miitato  ordine  mutant  naturam. 


IV]  STRUCTURE   OF   THE   CELL  161 

But  we  are  by  no  means  necessarily  in  this  dilemma.  For 
though  we  come  perilously  near  to  it  when  we  contemplate  the 
lowest  orders  of  magnitude  to  which  life  has  been  attributed,  yet 
in  the  case  of  the  ordinary  cell,  or  ordinary  egg  or  germ  which  is 
going  to  develop  into  a  complex  organism,  if  we  have  no  reason 
to  assume  or  to  believe  that  it  comprises  an  intricate  "  mechanism," 
we  may  be  quite  sure,  both  on  direct  and  indirect  evidence,  that, 
like  the  powder  in  our  rocket,  it  is  very  heterogeneous  in  its 
structure.  It  is  a  mixture  of  substances  of  various  kinds,  more 
or  less  fluid,  more  or  less  mobile,  influenced  in  various  ways  by 
chemical,  electrical,  osmotic,  and  other  forces,  and  in  their 
admixture  separated  by  a  multitude  of  surfaces,  or  boundaries,  at 
which  these,  or  certain  of  these  forces  are  made  manifest. 

Indeed,  such  an  arrangement  as  this  is  already  enough  to 
constitute  a  "mechanism";  for  we  must  be  very  careful  not  to 
let  our  physical  or  physiological  concept  of  mechanism  be  narrowed 
to  an  interpretation  of  the  term  derived  from  the  delicate  and 
complicated  contrivances  of  human  skill.  From  the  physical 
point  of  view,  we  understand  by  a  "mechanism"  whatsoever 
checks  or  controls,  and  guides  into  determinate  paths,  the  workings 
of  energy;  in  other  words,  whatsoever  leads  in  the  degradation 
of  energy  to  its  manifestation  in  some  determinate  form  of  ivorh, 
at  a  stage  short  of  that  ultimate  degradation  which  lapses  in 
uniformly  diffused  heat.  This,  as  Warburg  has  well  explained,  is 
the  general  eflect  or  function  of  the  physiological  machine,  and  in 
particular  of  that  part  of  it  which  we  call  "cell-structure*." 
The  normal  muscle-cell  is  something  which  turns  energy,  derived 
from  oxidation,  into  work;  it  is  a  mechanism  which  arrests  and 
utiUses  the  chemical  energy  of  oxidation  in  its  downward  course ; 
but  the  same  cell  when  injured  or  disintegrated,  loses  its  "use- 
fulness," and  sets  free  a  greatly  increased  proportion  of  its  energy 
in  the  form  of  heat. 

But  very  great  and  wonderful  things  are  done  after  this  manner 
by  means  of  a  mechanism  (whether  natural  or  artificial)  of 
extreme  simplicity.     A  pool  of  water,  by  virtue  of  its  surface, 

*  Otto  Warburg,  Beitrage  zur  Physiologie  der  Zelle,  insbesondere  iiber  die 
Oxidationsgeschwindigkeit  in  Zellen;  in  Asher-Spiro's  Ergebnisse  der  Physiologie. 
•XIV,  pp.  253-337,  1914  (see  p.  315).     (Cf.  Bayliss,  General  Physiology,  1915,  p.  590). 

T.  G.  li 


162  ON   THE   INTERNAL  FORM  AND  [ch. 

is  an  admirable  mechanism  for  the  making  of  waves ;  with  a  lump 
of  ice  in  it,  it  becomes  an  efficient  and  self-contained  mechanism 
for  the  making  of  currents.  The  great  cosmic  mechanisms  are 
stupendous  in  their  simplicity ;  and,  in  point  of  fact,  every  great 
or  little  aggregate  of  heterogeneous  matter  (not  identical  in 
'phase")  involves,  ipso  facto,  the  essentials  of  a  mechanisna. 
Even  a  non-living  colloid,  from  its  intrinsic  heterogeneity,  is  in 
this  sense  a  mechanism,  and  one  in  which  energy  is  manifested 
in  the  movement  and  ceaseless  rearrangement  of  the  constituent 
particles.  For  this  reason  Graham  (if  I  remember  rightly)  speaks 
somewhere  or  other  of  the  colloid  state  as  "the  dynamic  state  of 
matter";  or  in  the  same  philosopher's  phrase  (of  which  Mr 
Hardy*  has  lately  reminded  us),  it  possesses  "ewer^rmf." 

Let  us  turn  then  to  consider,  briefly  and  diagrammatically,  the 
structure  of  the  cell,  a  fertilised  germ-cell  or  ovum  for  instance, 
not  in  any  vain  attempt  to  correlate  this  structure  with  the 
structure  or  properties  of  the  resulting  and  yet  distant  organism ; 
but  merely  to  see  how  far,  by  the  study  of  its  form  and  its  changing 
internal  configuration,  we  may  throw  light  on  certain  forces  which 
are  for  the  time  being  at  work  within  it. 

We  may  say  at  once  that  we  can  scarcely  hope  to  learn  more 
of  these  forces,  in  the  first  instance,  than  a  few  facts  regarding 
their  direction  and  magnitude ;  the  nature  and  specific  identity 
of  the  force  or  forces  is  a  very  different  matter.  This  latter 
problem  is  likely  to  be  very  difficult  of  elucidation,  for  the  reason, 
among  others,  that  very  different  forces  are  often  very  much  alike 
in  their  outward  and  visible  manifestations.  So  it  has  come  to 
pass  that  we  have  a  multitude  of  discordant  hypotheses  as  to  the 
nature  of  the  forces  acting  within  the  cell,  and  producing,  in  cell 
division,  the  "  caryokinetic "  figures  of  which  we  are  about  to 
speak.  One  student  may,  like  Rhumbler,  choose  to  account  for 
them  by  an  hypothesis  of  mechanical  traction,  acting  on  a  reticular 
web  of  protoplasm  J ;  another,  like  Leduc,  may  shew  us  how  in 

*  Hardy,  W.  B.,  On  some  Problems  of  Living  Matter  (Guthrie  Lecture), 
Tr.  Physical  Soc.  London,  xxviii,  p.  99-118,  1916. 

I  As  a  matter  of  fact  both  phrases  occur,  side  by  side,  in  Graham's  classical 
paper  on  "Liquid  Diffusion  apphed  to  Analysis,"  Phil.  Trans,  cli,  p.  184,  1861; 
Chem.  and  Phys.  Researches  (ed.  Angus  Smith),  1876,  p.  554. 

X  L.   Rhumbler,   Mechanische    Erklarung    der  Aehnhchkeit   zwischen  Magne- 


IV]  STRUCTURE   OF   THE   CELL  163 

many  of  their  most  striking  features  they  may  be  admirably 
simulated  by  the  diffusion  of  salts  in  a  colloid  medium;  others 
again,  like  Gallardo*  and  Hartog,  and  Rhumbler  (in  his  earher 
papers)  f,  insist  on  their  resemblance  to  the  phenomena  of 
electricity  and  magnetism  J  ;  while  Hartog  beUeves  that  the  force 
in  question  is  only  analogous  to  these,  and  has  a  specific  identity 
of  its  own  §.  All  these  conflicting  views  are  of  secondary  import- 
ance, so  long  as  we  seek  only  to  account  for  certain  configurations 
which  reveal  the  direction,  rather  than  the  nature,  of  a  force. 
One  and  the  same  system  of  Hues  of  force  may  appear  in  a  field 
of  magnetic  or  of  electrical  energy,  of  the  osmotic  energy  of 
diffusion,  of  the  gravitational  energy  of  a  flowing  stream.  In  short, 
we  may  expect  to  learn  something  of  the  pure  or  abstract  dynamics, 
long  before  we  can  deal  with  the  special  physics  of  the  cell.  For 
indeed  (as  Maillard  has  suggested),  just  as  uniform  expansion 
about  a  single  centre,  to  whatsoever  physical  cause  it  may  be  due 
will  lead  to  the  configuration  of  a  sphere,  so  will  any  two  centres 
or  foci  of  potential  (of  whatsoever  kind)  lead  to  the  configurations 
with  which  Faraday  made  us  familiar  under  the  name  of  "lines 
of  force||";   and  this  is  as  much  as  to  say  that  the  phenomenon, 

tischen  Kraftliniensystemen  unci  Zelltheilungsfiguren,  Arch.  f.  Entw.  Mech.  xv, 
p.  482,  1903. 

*  Gallardo,  A.,  Essai  d'interijretation  des  figures  caryocinetiques,  Anales  del 
Miiseo  de  Buenos-Aires  (2),  ii,  1896;  La  division  de  la  cellule,  phenoraene  bipolaire 
de  caractere  electro-colloidal.  Arch.  f.  Entw.  Mech.  xxviii,  1909,  etc. 

t  Arch.  f.  Entw.  Mech.  m,  iv,  1896-97. 

J  On  various  theories  of  the  mechanism  of  mitosis,  see  (e.g.)  Wilson,  The  Cell 
in  Development,  etc.,  pp.  100-114;  Meves,  Zelltheilung,  in  Merkel  u.  Bonnet's 
Ergebnisse  der  Anatomic,  etc.,  vii,  viii,  1897-8;  Ida  H.  Hyde,  Amer.  Journ. 
of  Physiol.  XII,  pp.  241-275,  1905;  and  especially  Prenant,  A.,  Theories  et  inter- 
pretations physiques  de  la  mitose,  J.  de  VAnat.  et  Physiol,  xlvi,  pp.  511-578,  1910. 

§  Hartog,  M.,  Une  force  nouvelle:  le  mitokinetisme,  C.R.  11  Juli,  1910; 
Mitokinetism  in  the  Mitotic  Spindle  and  in  the  Polyasters,  Arch.  f.  Entw.  Mech. 
xxvn,  pp.  141-145,  1909;  cf.  ibid.  XL,  pp.  33-64,  1914.  Cf.  also  Hartog's  papers 
in  Proc.  R.  S.  (B),  lxxvi,  1905;  Science  Progress  (n.  s.),  i,  1907;  Riv.  di  Scienza, 
II,  1908;    C.  R.  Assoc.fr.  pour  VAvancem.  des  Sc.  1914,  etc. 

II  The  configurations,  as  obtained  by  the  usual  experimental  methods,  were 
of  course  known  long  before  Faraday's  day,  and  constituted  the  "convergent  and 
divergent  magnetic  curves"  of  eighteenth  century  mathematicians.  As  LesUe 
said,  in  1821,  they  were  "regarded  with  wonder  by  a  certain  class  of  dreaming 
philosophers,  who  did  not  hesitate  to  consider  them  as  the  actual  traces  of  an 
invisible  fluid,  perpetually  circulating  between  the  poles  of  the  magnet."  Faraday's 
great  advance  was  to  interpret  them    as  indications  of  stress  in  a  medium, — of 

11—2 


164  ON   THE   INTERNAL  FORM  AND  [ch. 

though  physical  in  the  concrete,  is  in  the  abstract  purely  mathe- 
matical, and  in  its  very  essence  is  neither  more  nor  less  than  a 
property  of  three-dimensional  space. 

But  as  a  matter  of  fact,  in  this  instance,  that  is  to  say  in 
trying  to  explain  the  leading  phenomena  of  the  caryokinetic 
division  of  the  cell,  we  shall  soon  perceive  that  any  explanation 
which  is  based,  like  Rhumbler's,  on  mere  mechanical  traction,  is 
obviously  inadequate,  and  we  shall  find  ourselves  limited  to  the 
hypothesis  of  some  polarised  and  polarising  force,  such  as  we  deal 
with,  for  instance,  in  the  phenomena  of  magnetism  or  electricity. 

Let  us  speak  first  of  the  cell  itself,  as  it  appears  in  a  state  of 
rest,  and  let  us  proceed  afterwards  to  study  the  more  active 
phenomena  which  accompany  its  division. 

Our  typical  cell  is  a  spherical  body ;  that  is  to  say,  the  uniform 
surface-tension  at  its  boundary  is  balanced  by  the  outward 
resistance  of  uniform  forces  within.  But  at  times  the  surface- 
tension  may  be  a  fluctuating  quantity,  as  when  it  produces  the 
rhythmical  contractions  or  "Ransom's  waves"  on  the  surface  of 
a  trout's  egg;  or  again,  while  the  egg  is  in  contact  with  other 
bodies,  the  surface-tension  may  be  locally  unequal  and  variable, 
giving  rise  to  an  amoeboid  figure,  as  in  the  egg  of  Hydra*. 

Within  the  ovum  is  a  nucleus  or  germinal  vesicle,  also  spherical, 

and  consisting  as  a  rule  of  portions  of  "chromatin,"  aggregated 

together  within  a   more  fluid  drop.     The  fact  has   often   been 

commented  upon  that,  in  cells  generally,  there  is  no  correlation 

of  form  (though  there  apparently  is  of  size)  between  the  nucleus 

and  the  "cytoplasm,"  or  main  body  of  the  cell.      So  Whitman "f 

remarks  that  "except  during  the  process  of  division  the  nucleus 

seldom  departs  from  its  typical  spherical  form.     It  divides  and 

sub-divides,  ever  returning  to  the  same  round  or  oval  form.... 

How  different  with  the  cell.     It  preserves  the  spherical  form  as 

rarely  as  the  nucleus  departs  from  it.     Variation  in  form  marks 

the   beginning  and   the   end   of  every  important  chapter  in  its 

tension  or  attraction  along  the  lines,  and  of  repulsion  transverse  to  the  hnes,  of  tlie 
diagram. 

*  Cf.  also  the  curious  phenomenon  in  a  dividing  egg  described  as  "spinning" 
by  Mrs  G.  F.  Andrews,  J.  of  Morph.  xn,  pp.  367-389,  1897. 

t  Whitman,  J.  of  Morph.  n,  p.  40,  1889. 


J 


IV]  STRUCTURE   OF   THE   CELL  165 

history."  On  simple  dynamical  gromids,  the  contrast  is  easily 
explained.  So  long  as  the  fluid  substance  of  the  nucleus  is  quah- 
tatively  different  from,  and  incapable  of  mixing  with,  the  fluid 
or  semi-fluid  protoplasm  which  surrounds  it,  we  shall  expect  it 
to  be,  as  it  almost  always  is,  of  spherical  form.  For,  on  the  one 
hand,  it  is  bounded  by  a  liquid  film,  whose  surface-tension  is 
uniform ;  and  on  the  other,  it  is  immersed  in  a  medium  which 
transmits  on  all  sides  a  uniform  fluid  pressure  *.  For  a  similar 
reason  the  contractile  vacuole  of  a  Protozoon  is  spherical  in  form : 
it  is  just  a  "drop"  of  fluid,  bounded  by  a  uniform  surface- 
tension  and  through  whose  boundary-film  diffusion  is  taking  place. 
But  here,  owing  to  the  small  difference  between  the  fluid  constitut- 
ing, and  that  surrounding,  the  drop,  the  surface-tension  equi- 
librium is  unstable ;  it  is  apt  to  vanish,  and  the  rounded  outline 
of  the  drop,  like  a  burst  bubble,  disappears  in  a  moment  t- 
The  case  of  the  spherical  nucleus  is  closely  akin  to  the  spherical 
form  of  the  yolk  within  the  bird's  egg  J.  But  if  the  substance  of 
the  cell  acquire  a  greater  solidity,  as  for  instance  in  a  muscle 

*  "Souvent  il  n'y  a  qu'une  separation  physique  entre  le  cytoplasme  et  le  sue 
nucleaire,  comme  entre  deux  liquides  immiscibles,  etc.;"  Alexeieff,  Sur  la  mitosc 
dite  ''primitive,"  Arch.  f.  Protistenk.  xxix,  p.  357,  1913. 

f  The  aj^pearance  of  "  vacuolation "  is  a  result  of  endosmosis  or  the  diffusion 
of  a  less  dense  fluid  into  the  denser  plasma  of  the  cell.  Caeferis  paribus,  it  is  less 
apparent  in  marine  organisms  than  in  those  of  freshwater,  and  in  many  or  most 
marine  Ciliates  and  even  Rhizopods  a  contractile  vacuole  has  not  been  observed 
(Biitschli,  in  Bronn's  Protozoa,  p.  1414) ;  it  is  also  absent,  and  probably  for  the  same 
reason,  in  parasitic  Protozoa,  such  as  the  Gregarines  and  the  Entamoebae.  Rossbach 
shewed  that  the  contractile  vacuole  of  ordinary  freshwater  Ciliates  was  very  greatly 
diminished  in  a  5  per  cent,  solution  of  NaCl,  and  all  but  disappeared  in  a  1  per  cent, 
solution  of  sugar  {Arb.  z.  z.  Inst.  Wurzburg,  1872,  of.  Massart,  Arch,  de  Biol,  lx, 
p.  515,  1889).  Actmophrys  sol,  when  gradually  acclimatised  to  sea-water,  loses  its 
vacuoles,  and  vice  versa  (Gruber,  Biol.  Centralbl.  ix,  p.  22,  1889) ;  and  the  same  is 
true  of  Amoeba  (Zuelzer,  Arch.  f.  Entw.  Mech.  1910,  p.  632).  The  gradual  enlarge- 
ment of  the  contractile  vacuole  is  precisely  analogous  to  the  change  of  size  of  a 
bubble  until  the  gases  on  either  side  of  the  film  are  equally  diffused,  as  described 
long  ago  by  Draper  {Phil.  Mag.  (n.  s.),  xi,  p.  559,  1837).  Rhumbler  has  shewn 
that  contractile  or  pulsating  vacuoles  may  be  well  imitated  in  chloroform-drops, 
suspended  in  water  in  which  various  substances  are  dissolved  {Arch.  f.  Entw. 
Mech.  VII,  1898,  p.  103).  The  pressure  within  the  contractile  vacuole,  always 
greater  than  without,  diminishes  with  its  size,  being  inversely  proportional  to 
its  radius;  and  when  it  lies  near  the  surface  of  the  cell,  as  in  a  Heliozoon,  it 
bursts  as  soon  as  it  reaches  a  thinness  which  its  viscosity  or  molecular  cohesion  no 
longer  permits  it  to  maintain. 

t  Cf,  p.  660. 


166  ON   THE   INTERNAL  FORM  AND  [ch. 

cell,  or  by  reason  of  mucous  accumulations  in  an  epithelium  cell, 
then  the  laws  of  fluid  pressure  no  longer  apply,  the  external 
pressure  on  the  nucleus  tends  to  become  unsymmetrical,  and  its 
shape  is  modified  accordingly.  "Amoeboid"  movements  may  be 
set  up  in  the  nucleus  by  anything  which  disturbs  the  symmetry  of 
its  own  surface-tension.  And  the  cases,  as  in  many  Rhizopods, 
where  "nuclear  material"  is  scattered  in  small  portions  throughout 
the  cell  instead  of  being  aggregated  in  a  single  nucleus,  are  probably 
capable  of  very  simple  explanation  by  supposing  that  the  "phase 
difEerence"  (as  the  chemists  say)  between  the  nuclear  and  the 
protoplasmic  substance  is  comparatively  slight,  and  the  surface- 
tension  which  tends  to  keep  them  separate  is  correspondingly 
small*. 

It  has  been  shewn  that  ordinary  nuclei,  isolated  in  a  living 
or  fresh  state,  easily  flow  together;  and  this  fact  is  enough  to 
suggest  that  they  are  aggregations  of  a  particular  substance  rather 
than  bodies  deserving  the  name  of  particular  organs.  It  is  by 
reason  of  the  same  tendency  to  confluence  or  aggregation  of 
particles  that  the  ordinary  nucleus  is  itself  formed,  until  the 
imposition  of  a  new  force  leads  to  its  disruption. 

Apart  from  that  invisible  or  ultra-microscopic  heterogeneity 
which  is  inseparable  from  our  notion  of  a  "colloid,"  there  is  a 
visible  heterogeneity  of  structure  within  both  the  nucleus  and  the 
outer  protoplasm.  The  former,  for  instance,  contains  a  rounded 
nucleolus  or  "germinal  spot,"  certain  conspicuous  granules  or 
strands  of  the  pecuHar  substance  called  chromatin,  and  a  coarse 
meshwork  of  a  protoplasmic  material  known  as  "Hnin"  or  achro- 
matin;  the  outer  protoplasm,  or  cytoplasm,  is  generally  believed 
to  consist  throughout  of  a  sponger  work,  or  rather  alveolar  mesh- 
work, of  more  and  less  fluid  substances ;  and  lastly,  there  are 
generally  to  be  detected  one  or  more  very  minute  bodies,  usually 
in  the  cytoplasm,  sometimes  within  the  nucleus,  known  as  the 
centrosome  or  centrosomes. 

The  morphologist  is  accustomed  to  speak  of  a  "polarity"  of 

*  The  elongated  or  curved  "  macronucleus "  of  an  Infusorian  is  to  be  looked 
upon  as  a  single  mass  of  chromatin,  rather  than  as  an  aggregation  of  particles  in 
a  fluid  drop,  as  in  the  case  described.  It  has  a  shape  of  its  own,  in  which  ordinary 
surface-tension  plays  a  very  subordinate  part. 


IV]  STRUCTURE   OF   THE   CELL  167 

the  cell,  meaning  thereby  a  symmetry  of  visible  structure  about 
a  particular  axis.  For  instance,  whenever  we  can  recognise  in 
a  cell  both  a  nucleus  and  a  centrosome,  we  may  consider  a 
line  drawn  through  the  two  as  the  morphological  axis  of  polarity : 
in  an  epithehum  cell,  it  is  obvious  that  the  cell  is  morphologically 
symmetrical  about  a  median  axis  passing  from  its  free  surface  to 
its  attached  base.  Again,  by  an  extension  of  the  term  "polarity," 
as  is  customary  in  dynamics,  we  may  have  a  "radial"  polarity, 
between  centre  and  periphery;  and  lastly,  we  may  have  several 
apparently  independent  centres  of  polarity  within  the  single  cell. 
Only  in  cells  of  quite  irregular,  or  amoeboid  form,  do  we  fail  to 
recognise  a  definite  and  symmetrical  "polarity."  The  morpho- 
logical "polarity"  is  accompanied  by,  and  is  but  the  outward 
expression  (or  part  of  it)  of  a  true  dynamical  polarity,  or  distribution 
of  forces;  and  the  "hues  of  force"  are  rendered  visible  by  con- 
catenation of  particles  of  matter,  such  as  come  under  the  influence 
of  the  forces  in  action. 

When  the  lines  of  force  stream  inwards  from  the  periphery 
towards  a  point  in  the  interior  of  the  cell,  the  particles  susceptible 
of  attraction  either  crowd  towards  the  surface  of  the  cell,  or,  when 
retarded  by  friction,  are  seen  forming  lines  or  "fibrillae"  which 
radiate  outwards  from  the  centre  and  constitute  a  so-called 
"aster."  In  the  cells  of  columnar  or  ciliated  epithelium,  where 
the  sides  of  the  cell  are  symmetrically  disposed  to  their  neighbours 
but  the  free  and  attached  surfaces  are  very  diverse  from  one 
another  in  their  external  relations,  it  is  these  latter  surfaces  which 
constitute  the  opposite  poles;  and  in  accordance  with  the  parallel 
lines  of  force  so  set  up,  we  very  frequently  see  parallel  lines  of 
granules  which  have  ranged  themselves  perpendicularly  to  the 
free  surface  of  the  cell  (cf.  fig.  97). 

A  simple  manifestation  of  "polarity"  may  be  well  illustrated 
by  the  phenomenon  of  diffusion,  where  w^e  may  conceive,  and  may 
automatically  reproduce,  a  "field  of  force,"  with  its  poles  and 
visible  lines  of  equipotential,  very  much  as  in  Faraday's  conception 
of  the  field  of  force  of  a  magnetic  system.  Thus,  in  one  of  Leduc's 
experiments*,  if  we  spread  a  layer  of  salt  solution  over  a  level 

*  Theorie  physico-chimique  de  la  Vie,  p.  73,  1910;  Mechanism  of  Life,  p.  56, 
1911. 


168  ON   THE  INTERNAL  FORM  AND  [ch. 

plate  of  glass,  and  let  fall  into  the  middle  of  it  a  drop  of  indian 
ink,  or  of  blood,  we  shall  find  the  coloured  particles  travelUng 
outwards  from  the  central  "pole  of  concentration  "  along  the  lines 
of  diffusive  force,  and  so  mapping  out  for  us  a  "monopolar  field" 
of  diffusion :  and  if  we  set  two  such  drops  side  by  side,  their 
lines  of  diffusion  will  oppose,  and  repel,  one  another.  Or,  instead 
of  the  uniform  layer  of  salt  solution,  we  may  place  at  a  little 
distance  from  one  another  a  grain  of  salt  and  a  drop  of  blood, 
representing  two  opposite  poles :  and  so  obtain  a  picture  of  a 
"bipolar  field"  of  diffusion.  In  either  case,  we  obtain  results 
closely  analogous  to  the  "morphological,"  but  really  dynamical, 
polarity  of  the  organic  cell.  But  in  all  probabiUty,  the  dynamical 
polarity,  or  asymmetry  of  the  cell  is  a  very  complicated  phenome- 
non :  for  the  obvious  reason  that,  in  any  system,  one  asymmetry 
^^'i^  tend  to  beget  another.  A  chemical  asymmetry  will  induce  an 
inequahty  of  surface-tension,  which  will  lead  directly  to  a  modifi- 
cation of  form ;  the  chemical  asymmetry  may  in  turn  be  due  to  a 
process  of  electrolysis  in  a  polarised  electrical  field;  and  again 
the  chemical  heterogeneity  may  be  intensified  into  a  chemical 
"polarity,"  by  the  tendency  of  certain  substances  to  seek  a  locus 
of  greater  or  less  surface-energy.  We  need  not  attempt  to 
grapple  with  a  subject  so  complicated,  and  leading  to  so  many 
problems  which  lie  beyond  the  sphere  of  interest  of  the  morph- 
ologist.  But  yet  the  morphologist,  in  his  study  of  the  cell, 
cannot  quite  evade  these  important  issues ;  and  we  shall  return 
to  them  again  when  we  have  dealt  somewhat  with  the  form  of 
the  cell,  and  have  taken  account  of  some  of  the  simpler  pheno- 
mena of  surface-tension. 

We  are  now  ready,  and  in  some  measure  prepared,  to  study 
the  numerous  and  complex  phenomena  which  usually  accompany 
the  division  of  the  cell,  for  instance  of  the  fertilised  egg. 

Division  of  the  cell  is  essentially  accompanied,  and  preceded, 
by  a  change  from  radial  or  monopolar  to  a  definitely  bipolar 
polarity. 

In  the  hitherto  quiescent,  or  apparently  quiescent  cell,  we  per- 
ceive certain  movements,  which  correspond  precisely  to  what  must 
accompany  and  result  from  a  "polarisation"  of  forces  within  the 


IV]  STRUCTURE   OF   THE   CELL  169 

cell :  of  forces  which,  whatever  may  be  their  specific  nature,  at  least 
are  capable  of  polarisation,  and  of  producing  consequent  attraction 
or  repulsion  between  charged  particles  of  matter.  The  opposing 
forces  which  were  distributed  in  equilibrium  throughout  the  sub- 
stance of  the  cell  become  focussed  at  two  "centrosomes,"  which 
may  or  may  not  be  already  distinguished  as'  visible  portions  of 
matter ;  in  the  egg,  one  of  these  is  always  near  to,  and  the  other 
remote  from,  the  "animal  pole"  of  the  egg,  which  pole  is  visibly 
as  well  as  chemically  different  from  the  other,  and  is  the  region  in 
which  the  more  rapid  and  conspicuous  developmental  changes  will 
presently  begin.     Between  the  two  centrosomes,  a  spindle-shaped 


Fig.   41.     Caryokinetic  figure  in  a  dividing  cell  (or  blastomere)  of  the  Trout's 
egg.     (After  Prenant,  from  a  preparation  by  Prof.   P.  Bouin. ) 

figure  appears,  whose  striking  resemblance  to  the  lines  of  force 
made  visible  by  iron-filings  between  the  poles  of  a  magnet,  was  at 
once  recognised  by  Hermann  Fol,  when  in  1873  he  witnessed  for 
the  first  time  the  phenomenon  in  question.  On  the  farther  side 
of  the  centrosomes  are  seen  star-like  figures,  or  "asters,"  in  which 
we  can  without  difficulty  recognise  the  broken  lines  of  force  which 
run  externally  to  those  stronger  lines  which  lie  nearer  to  the  polar 
axis  and  which  constitute  the  "spindle."  The  lines  of  force  are 
rendered  visible  or  "material,"  just  as  in  the  experiment  of  the 
iron-fil  ngs,  by  the  fact  that,  in  the  heterogeneous  substance  of 
the  cell,  certain  portions  of  matter  are  more  "permeable"  to  the 
acting  force  than  the  rest,  become  themselves  polarised  after  the 


170  ON   THE   INTERNAL  FORM  AND  [ch. 

fashion  of  a  magnetic  or  "paramagnetic"  body,  arrange  themselves 
in  an  orderly  way  between  the  two  poles  of  the  field  of  force,  cling 
to  one  another  as  it  were  in  threads*,  and  are  only  prevented  by 
the  friction  of  the  surrounding  medium  from  approaching  and 
congregating  around  the  adjacent  poles. 

As  the  field  of  force  strengthens,  the  more  will  the  hnes  of  force 
be  drawn  in  towards  the  interpolar  axis,  and  the  less  evident  will 
be  those  remoter  lines  which  constitute  the  terminal,  or  extrapolar, 
asters :  a  clear  space,  free  from  materialised  lines  of  force,  may 
thus  tend  to  be  set  up  on  either  side  of  the  spindle,  the 
so-called  "Biitschli  space"  of  the  histologistsf.  On  the  other 
hand,  the  lines  of  force  constituting  the  spindle  will  be  less  con- 
centrated if  they  find  a  path  of  less  resistance  at  the  periphery 
of  the  cell :  as  happens,  in  our  experiment  of  the  iron-filings,  when 
we  encircle  the  field  of  force  with  an  iron  ring.  On  this  principle, 
the  differences  observed  between  cells  in  which  the  spindle  is  well 
developed  and  the  asters  small,  and  others  in  which  the  spindle 
is  weak  and  the  asters  enormously  developed,  can  be  easily 
explained  by  variations  in  the  potential  of  the  field,  the  large, 
conspicuous  asters  being  probably  correlated  with  a  marked 
permeability  of  the  surface  of  the  cell. 

The  visible  field  of  force,  though  often  called  the  "nuclear 
spindle,"  is  formed  outside  of,  but  usually  near  to,  the  nucleus. 
Let  us  look  a  httle  more  closely  into  the  structure  of  this  body, 
and  into  the  changes  which  it  presently  undergoes. 

Within  its  spherical  outhne  (Fig.  42),  it  contains  an  "alveolar" 

*  Whence  the  name  "mitosis"  (Greek  /j-ltos,  a  thread),  applied  first  by  Flemming 
to  the  whole  phenomenon.  Kollmann  (Biol.  Centralbl.  ii,  p.  107,  1882)  called  it 
divisio  per  fila,  or  divisio  laqueis  implicata.  Many  of  the  earher  students,  such  as 
Van  Beneden  (Rech.  sur  la  maturation  de  I'oeuf,  Arch,  de  Biol,  iv,  1883),  and 
Hermann  (Zur  Lehre  v.  d.  Entstehung  d.  karyokinetischen  Spindel,  Arch.f.  mikrosk. 
Anat.  XXXVII,  1891)  thought  they  recognised  actual  muscular  threads,  drawing 
the  nuclear  material  asunder  towards  the  respective  foci  or  poles;  and  some  such 
view  was  long  maintained  by  other  writers,  Boveri,  Heidenhain,  Flemming,  R. 
Hertwig,  and  many  more.  In  fact,  the  existence  of  contractile  threads,  or  the 
ascription  to  the  spindle  rather  than  to  the  poles  or  centrosomes  of.  the  active 
forces  concerned  in  nuclear  division,  formed  the  main  tenet  of  all  those  who  dechned 
to  go  beyond  the  "contractile  properties  of  protoplasm"  for  an  explanation  of  the 
phenomenon.     (Cf.  also  J.  W.  Jenkinson,  Q.  J.  M.  S.  xlviii,  p.  471,  1904.) 

f  Cf.  Biitschli,  0.,  Ueber  die  kiinsthche  Nachahmung  der  karj'okinetischen 
.Figur,  Verh.  Med.  Nat.  Ver.  Heidelberg,  v,  pp.  28-41  (1892),  1897. 


IV] 


STRUCTURE   OF   THE   CELL 


171 


mesh  work  (often  described,  from  its  appearance  in  optical  section, 
as  a  "reticulum"),  consisting  of  more  solid  substances,  with  more 
fluid  matter  filling  up  the  interalveolar  meshes.  This  phenomenon 
is  nothing  else  than  what  we  call  in  ordinary  language,  a  "froth"' 
or  a  "foam."  It  is  a  surface-tension  phenomenon,  due  to  the 
interacting  surface-tensions  of  two  intermixed  fluids,  not  very 
different  in  density,  as  they  strive  to  separate.  Of  precisely  the 
same  kind  (as  Biitschli  was  the  first  to  shew)  are  the  minute  alveolar 
networks  which  are  to  be  discerned  in  the  cytoplasm  of  the  cell*, 
and  which  we  now  know  to  be  not  inherent  in  the  nature  of 


Fig.  42. 


Fig.  43. 


protoplasm,  or  of  living  matter  in  general,  but  to  be  due  to  various 
causes,  natural  as  well  as  artificial.  The  microscopic  honeycomb 
structure  of  cast  metal  under  various  conditions  of  cooling,  even 
on  a  grand  scale  the  columnar  structure  of  basaltic  rock,  is  an 
example  of  the  same  surface-tension  phenomenon. 

*  Arrhenius,  in  describing  a  typical  colloid  precipitate,  does  so  in  terms  that 
are  very  closely  applicable  to  the  ordinary  microscopic  appearance  of  the  protojjlasm 
of  the  cell.  The  precipitate  consists,  he  says,  "en  un  reseau  d'mie  substance 
solide  contenant  peu  d'eau,  dans  les  mailles  duquel  est  inclus  un  fluide  contenant 
un  peu.de  coUoide  dans  beaucoup  d'eau. ..Evidemment  cette  structure  se  forme 
a  cause  de  la  petite  difference  de  poids  specifique  des  deux  phases,  et  de  la  con- 
sistance  gluante  des  particules  separees.  qui  s'attachent  en  forme  de  reseau."  Rev. 
Scientifique,  Feb.  1911. 


172  ON  THE   INTERNAL  FORM  AND  [ch. 

But  here  we  touch  the  brink  of  a  subject  so  important  that  we  must  not 
pass  it  by  without  a  word,  and  yet  so  contentious  that  we  must  not  enter  into 
its  details.  The  question  involved  is  simply  whether  the  great  mass  of 
recorded  observations  and  accepted  beliefs  with  regard  to  the  visible  structure 
of  protoplasm  and  of  the  cell  constitute  a  fair  picture  of  the  actual  living  cell, 
or  be  based  on  appearances  which  are  incident  to  death  itself  and  to  the 
artificial  treatment  which  the  microscopist  is  accustomed  to  apply.  The  great 
bulk  of  histological  work  is  done  by  methods  which  involve  the  sudden  killing 
of  the  cell  or  organism  by  strong  reagents,  the  assumption  being  that  death 
is  so  rapid  that  the  visible  phenomena  exhibited  diu'ing  life  are  retained  or 
"'fixed"'  in  our  preparations.  While  this  assumption  is  reasonable  and 
justified  as  regards  the  general  outward  form  of  small  organisms  or  of  individual 
cells,  enough  has  been  done  of  late  years  to  shew  that  the  case  is  totally 
different  in  the  case  of  the  minute  internal  networks,  granules,  etc.,  which 
represent  the  alleged  structure  of  protoplasm.  For,  as  Hardy  puts  it,  "It  is 
notorious  that  the  various  fixing  reagents  are  coagulants  of  organic  colloids, 
and  that  they  produce  precipitates  which  have  a  certain  figure  or  structure,... 
and  that  the  figure  varies,  other  things  being  equal,  according  to  the  reagent 
used."  So  it  comes  to  pass  that  some  writers*  have  altogether  denied  the 
existence  in  the  living  cell-protoplasm  of  a  network  or  alveolar  "foam"; 
others  t  have  cast  doubts  on  the  main  tenets  of  recent  histology  regarding 
nuclear  structure ;  and  Hardy,  discussing  the  structure  of  certain  gland-cells, 
declares  that  "there  is  no  evidence  that  the  structure  discoverable  in  the  cell- 
substance  of  these  cells  after  fixation  has  any  counterpart  in  the  cell  when 
living."  "A  large  part  of  it  "  he  goes  on  to  say  "is  an  artefact.  The 
profound  difference  in  the  minute  structure  of  a  secretory  cell  of  a  mucous 
gland  according  to  the  reagent  which  is  used  to  fix  it  would,  it  seems 
to  me,  almost  suffice  to  establish  this  statement  in  the  absence  of  other 
evidence." 

Nevertheless,  histological  study  proceeds,  especially  on  the  part  of  the 
morphologists,  with  but  little  change  in  theory  or  in  method,  in  spite  of  these 
and  many  other  warnings.  That  certain  visible  sti'uctures,  nucleus,  vacuoles, 
"attraction-spheres"  or  centrosomes,  etc.,  are  actually  present  in  the  living 
cell,  we  know  for  certain;  and  to  this  class  belong  the  great  majority  of 
structures  (including  the  nuclear  "spindle"  itself)  with  which  we  are  at  present 
concerned.  That  many  other  alleged  structures  are  artificial  has  also  been 
placed  beyond  a  doubt ;  but  where  to  draw  the  dividing  line  we  often  do  not 
know  j. 

*  F.  Schwartz,  in  Cohn's  Beifr.  z.  Biologie  der  Pflanzen,  v,  p.  1,  1887. 

t  Fischer,  Anat.  Anzeiger,  ix,  p.  678,  1894,  x,  p.  769,  1895. 

J  See,  in  particular,  W.  B.  Hardy,  On  the  structure  of  Cell  Protoplasm,  Journ. 
of  Physiol.  XXIV,  pp.  158-207,  1889;  also  Hober,  Physikalische  Chemie  der  Zelle 
uiid  der  Gewebe.  1902.  Cf.  (int.  al.)  Flemming,  Zellsubstanz,  Kern  und  Zelltheilung 
1882,  p.  51,  etc. 


IV] 


STRUCTURE   OF   THE   CELL 


173 


The  following  is  a  brief  epitome  of  the  visible  changes  undergone 
by  a  typical  cell,  leading  up  to  the  act  of  segmentation,  and  con- 
stituting the  phenomenon  of  mitosis  or  caryokinetic  division.  In 
the  egg  of  a  sea-urchin,  we  see  with  almost  diagrammatic  com- 
pleteness what  is  set  forth  here*. 

1.  The  chromatin,  which  to  begin  with  was  distributed  in 
granules  on  the  otherwise  achromatic  reticulum  (Fig.  42),  concen- 
trates to  form  a  skein  or  sfireme,  which  may  be  a  continuous 
thread  from  the  first  (Figs.  43,  44),  or  from  the  first  segmented. 
In  any  case  it  divides  transversely  sooner  or  later  into  a  number 
of  chromosomes  (Fig.  45),  which  as  a  rule  have  the  shape  of  little 


Fig.  44. 


rods,  straight  or  curved,  often  bent  into  a  V,  but  which  may 
also  be  ovoid,  or  round,  or  even  annular.  Certain  deeply  staining 
masses,  the  nucleoli,  which  may  be  present  in  the  resting  nucleus. 
do  not  take  part  in  the  process  of  chromosome  formation ;  they 
are  either  cast  out  of  the  nucleus  and  are  dissolved  in  the  cyto- 
plasm, or  fade  away  in  situ. 

2.  Meanwhile,  the  deeply  staining  granule  (here  extra - 
nuclear),  known  as  the  centrosome,  has  divided  in  two.  The  two 
resulting  granules  travel  to  opposite  poles  of  the  nucleus,   and 


*  My  description  and  diagrams  (Figs  42 — 51)  are  based  on  those  of  Professor 
E.  B.  WUson. 


174 


ON   THE   INTERNAL  EORM  AND 


[CH. 


there  each  becomes  surrounded  by  a  system  of  radiating  Unes,  the 
asters;  immediately  around  the  centrosome  is  a  clear  space,  the 
centrosphere  (Figs.  43-45).  Between  the  two  centrosomes  with 
their  asters  stretches  a  bundle  of  achromatic  fibres,  the  spindle. 

3.  The  surface-film  bounding  the  nucleus  has  broken  down, 
the  definite  nuclear  boundaries  are  lost,  and  the  spindle  now 
stretches  through  the  nuclear  material,  in  which  he  the  chromo- 
somes (Figs.  45,  46).  These  chromosomes  now  arrange  them- 
selves midway  between  the  poles  of  the  spindle,  where  they  form 
what  is  called  the  equatorial  plate  (Fig.  47). 

4.  Each  chromosome  splits  longitudinally  into  two :   usually 


equatorial  plate 

spindle  fibres 


Fig.  46. 


Fig.  47. 


at  this  stage, — but  it  is  to  be  noticed  that  the  splitting  may  have 
taken  place  so  early  as  the  spireme  stage  (Fig.  48). 

5.  The  halves  of  the  split  chromosomes  now  separate  from 
one  another,  and  travel  in  opposite  directions  towards  the  two 
poles  (Fig.  49).  As  they  move,  it  becomes  apparent  that  the  spindle 
consists  of  a  median  bundle  of  "fibres,"  the  central  spindle,  running 
from  pole  to  pole,  and  a  more  superficial  sheath  of  "mantle- 
fibres,"  to  which  the  chromosomes  seem  to  be  attached,  and  by 
which  they  seem  to  be  drawn  towards  the  asters. 

6.  The  daughter  chromosomes,  arranged  now  in  two  groups, 
become  closely  crowded  in  a  mass  near  the  centre  of  each  aster 


IV] 


STRUCTURE   OF  THE   CELL 


175 


(Fig.  50).    They  fuse  together  and  form  once  more  an  alveolar  reti- 
culum and  may  occasionally  at  this  stage  form  another  spireme. 


central  spindie 

manHe-fii\ 


cplit  chrcmosomc 2 

Fig.  48. 


Fis.  49. 


A  boundary  or  surface  wall  is  now  developed  round  each  recon- 
structed nuclear  mass,  and  the  spindle-fibres  disappear  (Fig.  51). 
The  centrosome  remains,  as  a  rule,  outside  the  nucleus. 


7.  On  the  central  spindle,  in  the  position  of  the  equatorial 
plate,  there  has  appeared  during  the  migration  of  the  chromosomes, 
a  "cell-plate"  of  deeply  staining  thickenings  (Figs.  50,  51).  This 
is  more  conspicuous  in  plant-cells. 


176  ON  THE   INTERNAL  FOEM  AND  [ch. 

8.  A  constriction  has  meanwhile  appeared  in  the  cytoplasm, 
and  the  cell  divides  through  the  equatorial  plane.  In  plant-cells 
the  line  of  this  division  is  foreshadowed  by  the  "cell-plate,"  which 
extends  from  the  spindle  across  the  entire  cell,  and  splits  into 
two  layers,  between  which  appears  the  membrane  by  which  the 
daughter  cells  are  cleft  asunder.  In  animal  cells  the  cell-plate 
does  not  attain  such  dimensions,  and  no  cell-wall  is  formed. 

The  whole,  or  very  nearly  the  whole  of  these  nuclear  phenomena 
may  be  brought  into  relation  with  that  polarisation  of  forces,  in 
the  cell  as  a  whole,  whose  field  is  made  manifest  by  the  "spindle" 
and  "asters"  of  which  we  have  already  spoken :  certain  particular 
phenomena,  directly  attributable  to  surface-tension  and  diffusion, 
taking  place  in  more  or  less  obvious  and  inevitable  dependence 
upon  the  polar  system*. 

At  the  same  time,  in  attempting  to  explain  the  phenomena,  we 
cannot  say  too  clearly,  or  too  often,  that  all  that  we  are  meanwhile 
justified  in  doing  is  to  try  to  shew  that  such  and  such  actions  lie 
within  the  range  of  known  physical  actions  and  phenomena,  or  that 
known  physical  phenomena  produce  effects  similar  to  them.  We 
want  to  feel  sure  that  the  whole  phenomenon  is  not  sui  generis,  but 
is  somehow  or  other  capable  of  being  referred  to  dynamical  laws, 
and  to  the  general  principles  of  physical  science.  But  when  we 
speak  of  some  particular  force  or  mode  of  action,  using  it  as  an 
illustrative  hypothesis,  we  must  stop  far  short  of  the  imphcation 
that  this  or  that  force  is  necessarily  the  very  one  which  is  actually 
at  work  within  the  living  cell ;  and  certainly  we  need  not  attempt 
the  formidable  task  of  trying  to  reconcile,  or  to  choose  between, 
the  various  hypotheses  which  have  already  been  enunciated,  or 
the  several  assumptions  on  which  they  depend. 

Any  region  of  space  within  which  action  is  manifested  is  a 
field  of  force ;  and  a  simple  example  is  a  bipolar  field,  in  which 
the  action  is  symmetrical  with  reference  to  the  line  joining  two 
points,  or  poles,  and  also  with  reference  to  the  "  equatorial " 
plane  equidistant  from  both.     We  have  such  a  "field  of  force"  in 

*  The  reference  numbers  in  the  following  account  refer  to  the  paragraphs  and 
figures  of  the  preceding  summary  of  visible  nuclear  phenomena. 


ivl  STRUCTURE   OF   THE   CELL  177 

the  neighbourhood  of  the  centrosome  of  the  ripe  cell  or  ovum, 
when  it  is  about  to  divide :  and  by  the  time  the  centrosome  has 
divided,  the  field  is  definitely  a  bipolar  one. 

The  quality  of  a  medium  filling  the  field  of  force  may  be  uniform, 
or  it  may  vary  from  point  to  point.  In  particular,  it  may  depend 
upon  the  magnitude  of  the  field ;  and  the  quality  of  one  medium 
may  differ  from  that  of  another.  Such  variation  of  quality, 
within  one  medium,  or  from  one  medium  to  another,  is  capable 
of  diagrammatic  representation  by  a  variation  of  the  direction  or 
the  strength  of  the  field  (other  conditions  being  the  same)  from  the 
state  manifested  in  some  uniform  medium  taken  as  a  standard. 
The  medium  is  said  to  be  permeable  to  the  force,  in  greater  or  less 
degree  than  the  standard  medium,  according  as  the  variation  of 
the  density  of  the  lines  of  force  from  the  standard  case,  under 
otherwise  identical  conditions,  is  in  excess  or  defect.  A  body 
placed  in  the  medium  will  tend  to  move  toivards  regions  of  greater  or 
less  force  according  as  its  'permeability  is  greater  or  less  than  that  oj 
the  surrounding  medium*.  In  the  common  experiment  of  placing 
iron-fiHngs  between  the  two  poles  of  a  magnetic  field,  the  fihngs 
have  a  very  high  permeability ;  and  not  only  do  they  themselves 
become  polarised  so  as  to  attract  one  another,  but  they  tend  to 
be  attracted  from  the  weaker  to  the  stronger  parts  of  the  field,  and 
as  we  have  seen,  were  it  not  for  friction  or  some  other  resistance, 
they  would  soon  gather  together  around  the  nearest  pole.  But 
if  we  repeat  the  same  experiment  with  such  a  metal  as  bismuth, 
which  is  very  little  permeable  to  the  magnetic  force,  then  the 
conditions  are  reversed,  and  the  particles,  being  repelled  from  the 
stronger  to  the  weaker  parts  of  the  field,  tend  to  take  up  their 
position  as  far  from  the  poles  as  possible.  The  particles  have 
become  polarised,  but  in  a  sense  opposite  to  that  of  the  surround- 
ing, or  adjacent,  field. 

Now,  in  the  field  of  force  whose  opposite  poles  are  marked  by 

*  If  the  word  permeability  be  deemed  too  directly  suggestive  of  the  phenomena 
of  magnetism  we  may  replace  it  by  the  more  general  term  of  specific  inductive 
capacity.  This  would  cover  the  particular  case,  which  is  by  no  means  an  improbable 
one,  of  our  phenomena  being  due  to  a  "'surface  charge"  borne  by  the  nucleus 
itself  and  also  by  the  chromosomes :  this  surface  charge  being  in  turn  the  result 
of  a  difference  in  inductive  capacity  between  the  body  or  particle  and  its  surrounding 
medium.     (Cf.  footnote,  p.  187.) 

T.  G.  12 


178  ON  THE   INTERNAL  FORM  AND  [ch. 

the  centrosonies  the  nucleus  appears  to  act  as  a  more  or  less  perme- 
able body,  as  a  body  more  permeable  than  the  surrounding  medium, 
that  is  to  say  the  "cytoplasm"  of  the  cell.  It  is  accordingly 
attracted  by,  and  drawn  into,  the  field  of  force,  and  tries,  as  it 
were,  to  set  itself  between  the  poles  and  as  far  as  possible  from 
both  of  them.  In  other  words,  the  centrosome-foci  will  be 
apparently  drawn  over  its  surface,  until  the  nucleus  as  a  whole 
is  involved  within  the  field  of  force,  which  is  visibly  marked  out 
by  the  "spindle"  (par.  3,  Figs.  44,  45). 

If  the  field  of  force  be  electrical,  or  act  in  a  fashion  analogous 
to  an  electrical  field,  the  charged  nucleus  will  have  its  surface- 
tensions  diminished  * :  with  the  double  result  that  the  inner 
alveolar  meshwork  will  be  broken  up  (par.  1),  and  that  the 
spherical  boundary  of  the  whole  nucleus  will  disappear  (par.  2). 
The  break-up  of  the  alveoh  (by  thinning  and  rupture  of  their 
partition  walls)  leads  to  the  formation  of  a  net,  and  the  further 
break-up  of  the  net  may  lead  to  the  unravelling  of  a  thread  or 
"spireme"  (Figs.  43,  44). 

Here  there  comes  into  play  a  fundamental  principle  which, 
in  so  far  as  we  require  to  understand  it,  can  be  explained  in  simple 
words.  The  effect  (and  we  might  even  say  the  object)  of  drawing 
the  more  permeable  body  in  between  the  poles,  is  to  obtain  an 
"easier  path"  by  which  the  lines  of  force  may  travel;  but  it  is 
obvious  that  a  longer  route  through  the  more  permeable  body 
may  at  length  be  found  less  advantageous  than  a  shorter  route 
through  the  less  permeable  medium.  That  is  to  say,  the  more 
permeable  body  will  only  tend  to  be  drawn  in  to  the  field  of  force 
until  a  point  is  reached  where  (so  to  speak)  the  way  round  and 
the  way  through  are  equally  advantageous.  We  should  accordingly 
expect  that  (on  our  hypothesis)  there  would  be  found  cases  in 
which  the  nucleus  was  wholly,  and  others  in  which  it  was  only 
partially,  and  in  greater  or  less  degree,  drawn  in  to  the  field 
between  the  centrosomes.  This  is  precisely  what  is  found  to 
occur  in  actual  fact.  Figs.  44  and  45  represent  two  so-called 
"types,"  of  a  phase  which  follows  that  represented  in  Fig.  43. 
According  to  the  usual  descriptions  (and  in  particular  to  Professor 

*  On  the  effect  of  electrical  influences  in  altering  the  surface-tensions  of  the 
colloid  particles,  see  Bredig,  Anorganische  Fermente,  pp.  15,  16,  1901. 


IV]  STRUCTURE   OF   THE   CELL  179 

E.  B.  Wilson's*),  we  are  told  that,  in  such  a  case  as  Fig.  44,  the 
"primary  spindle"  disappears  and  the  centrosomes  diverge  to 
opposite  poles  of  the  nucleus;  such  a  condition  being  found  in 
many  plant-cells,  and  in  the  cleavage-stages  of  many  eggs.  In 
Fig.  45,  on  the  other  hand,  the  primary  spindle  persists,  and 
subsequently  comes  to  form  the  main  or  "central"  spindle; 
while  at  the  same  time  we  see  the  fading  away  of  the  nuclear 
membrane,  the  breaking  up  of  the  spireme  into  separate  chromo- 
somes, and  an  ingrowth  into  the  nuclear  area  of  the  "astral  rays," 
— all  as  in  Fig.  46,  which  represents  the  next  succeeding  phase  of 
Fig.  45.  This  condition,  of  Fig.  46,  occurs  in  a  variety  of  cases; 
it  is  well  seen  in  the  epidermal  cells  of  the  salamander,  and  is 
also  on  the  whole  characteristic  of  the  mode  of  formation  of  the 
"polar  bodies."  It  is  clear  and  obvious  that  the  two  "types" 
correspond  to  mere  differences  of  degree,  and  are  such  as  would 
naturally  be  brought  about  by  differences  in  the  relative  per- 
meabilities of  the  nuclear  mass  and  of  the  surrounding  cytoplasm, 
or  even  by  differences  in  the  magnitude  of  the  former  body. 

But  now  an  important  change  takes  place,  or  rather  an 
important  difference  appears ;  for,  whereas  the  nucleus  as  a  whole 
tended  to  be  drawn  in  to  the  stronger  parts  of  the  field,  when  it 
comes  to  break  up  we  find,  on  the  contrary,  that  its  contained 
spireme-thread  or  separate  chromosomes  tend  to  be  repelled  to 
the  weaker  parts.  Whatever  this  difference  may  be  due  to, — 
whether,  for  instance,  to  actual  differences  of  permeabihty,  or 
possibly  to  differences  in  "surface-charge," — the  fact  is  that  the 
chromatin  substance  now  behaves  after  the  fashion  of  a  "dia- 
magnetic"  body,  and  is  repelled  from  the  stronger  to  the  weaker 
parts  of  the  field.  In  other  words,  its  particles,  lying  in  the 
inter-polar  field,  tend  to  travel  towards  the  equatorial  plane 
thereof  (Figs.  47,  48),  and  further  tend  to  move  outwards  towards 
the  periphery  of  that  plane,  towards  what  the  histologist 
calls  the  "mantle-fibres,"  or  outermost  of  the  lines  of  force  of 
which  the  spindle  is  made  up  (par.  5,  Fig.  47).  And  if  this  com- 
paratively non-permeable  chromatin  substance  come  to  consist  of 
separate  portions,  more  or  less  elongated  in  form,  these  portions, 
or  separate  "chromosomes,"  will  adjust  themselves  longitudinally, 

*  The  Cell,  etc.  p.  66. 

12—2 


180 


ON   THE   INTERNAL  FORM  AND 


[CH. 


in  a  peripheral  equatorial  circle  (Figs.  48,  49).  This  is  precisely 
what  actually  takes  place.  Moreover,  before  the  breaking  up  of 
the  nucleus,  long  before  the  chromatin  material  has  broken  up 
into  separate  chromosomes,  and  at  the  very  time  when  it  is  being 
fashioned  into  a  "spireme,"  this  body  already  lies  in  a  polar  field, 
and  must  already  have  a  tendency  to  set  itself  in  the  equatorial 
plane  thereof.  But  the  long,  continuous  spireme  thread  is  unable, 
so  long  as  the  nucleus  retains  its  spherical  boundary  wall,  to 
adjust  itself  in  a  simple  equatorial  annulus ;  in  striving  to  do  so, 
it  must  tend  to  coil  and  "kink"  itself,  and  in  so  doing  (if  all  this 
be  so),  it  must  tend  to  assume  the  characteristic  convolutions  of 
the  "spireme." 

After  the  spireme  has  broken  up  into  separate  chromosomes, 
these  particles  come  into  a  position  of  temporary,  and  unstable, 


Fig.  52.     Chromosomes,  undergoing  splitting  and  separation. 
(After  Hatschek  and  FJemming,  diagrammatised.) 

equilibrium  near  the  periphery  of  the  equatorial  plane,  and 
here  they  tend  to  place  themselves  in  a  symmetrical  arrange- 
ment (Fig.  52).  The  particles  are  rounded,  linear,  sometimes 
annular,  similar  in  form  and  size  to  one  another;  and 
lying  as  they  do  in  a  fluid,  and  subject  to  a  symmetrical  system 
of  forces,  it  is  not  surprising  that  they  arrange  themselves 
in  a  symmetrical  manner,  the  precise  arrangement  depending 
on  the  form  of  the  particles  themselves.  This  symmetry  may 
perhaps  be  due,  as  has  already  been  suggested,  to  induced 
electrical  charges.  In  discussing  Brauer's  observations  on  the 
splitting  of  the  chromatic  filament,  and  the  symmetrical  arrange- 
ment of  the  separate  granules,  in  Ascaris  megalocephala,  Lillie* 

*  Lillie,  R.  S.,  Amer.  J.  of  Physiol,  viii,  p.  282,   1903. 


IV] 


STRUCTURE   OF   THE   CELL 


181 


remarks:  "This  behaviour  is  strongly  suggestive  of  the  division 
of  a  colloidal  particle  under  the  influence  of  its  surface  electrical 
charge,  and  of  the  effects  of  mutual  repulsion  in  keeping  the 
products  of  division  apart."  It  is  also  probable  .that  surface- 
tensions  between  the  particles  and  the  surrounding  protoplasm 
would  bring  about  an  identical  result,  and  would  sufficiently 
account  for  the  obvious,  and  at  first  sight,  very  curious,  symmetry. 
We  know  that  if  we  float  a  couple  of  matches  in  water  they  tend 
to  approach  one  another,  till  they  lie  close  together,  side  by  side ; 
and,  if  we  lay  upon  a  smooth  wet  plate  four  matches,  half  broken 
across,  a  precisely  similar  attraction  brings  the  four  matches 
together  in  the  form  of  a  svmmetrical  cross.  Whether  one  of 
these,  or  some  other,  be  the  actual  explanation  of  the  phenomenon, 
it  is  at  least  plain  that  by  some  physical  cause,  some  mutual  and 


-Fig.  53.     Annular  chromosomes,  formed  in  the  spermatogenesis  of 
the  Mole-cricket,     (From  Wilson,  after  Vom  Rath.) 

symmetrical  attraction  or  repulsion  of  the  particles,  we  must  seek 
to  account  for  the  curious  symmetry  of  these  so-called  '"tetrads." 
The  remarkable  annular  chromosomes,  shewn  in  Fig.  53,  can  also 
be  easily  imitated  by  means  of  loops  of  thread  upon  a  soapy  film 
when  the  film  within  the  annulus  is  broken  or  its  tension  reduced. 


So  far  as  we  have  now  gone,  there  is  no  great  difficulty  in 
pointing  to  simple  and  familiar  phenomena  of  a  field  of  force 
which  are  similar,  or  comparable,  to  the  phenomena  which  we 
witness  within  the  cell.  But  among  these  latter  phenomena 
there  are  others  for  which  it  is  not  so  easy  to  suggest,  in  accordance 
with  known  laws,  a  simple  mode  of  physical  causation.  It  is  not 
at  once  obvious  how,  in  any  simple  system  of  symmetrical  forces, 


182 


ON   THE  INTERNAL  FORM  AND 


[CH. 


the  chromosomes,  which  had  at  first  been  apparently  repelled 
from  the  poles  towards  the  equatorial  plane,  should  then  be  spht 
asunder,  and  should  presently  be  attracted  in  opposite  directions, 
some  to  one  pole  and  some  to  the  other.  Remembering  that  it  is 
not  our  purpose  to  assert  that  some  one  particular  mode  of  action 
is  at  work,  but  merely  to  shew  that  there  do  exist  physical  forces, 
or  distributions  of  force,  which  are  capable  of  producing  the 
required  result,  I  give  the  following  suggestive  hypothesis,  which 
I  owe  to  my  colleague  Professor  W.  Peddie. 

As  we  have  begun  by  supposing  that  the  nuclear,  or  chromo- 
somal matter  differs  in  permeability  from  the  medium,  that  is  to 


Fb 


Fig.  54. 


say  the  cytoplasm,  in  which  it  Ues,  let  us  now  make  the  further 
assumption  that  its  permeabihty  is  variable,  and  depends  upon  the 
strength  of  the  field. 

In  Fig.  54,  we  have  a  field  of  force  (representing  our  cell), 
consisting  of  a  homogeneous  medium,  and  including  two  opposite 
poles :  fines  of  force  are  indicated  by  full  lines,  and  loci  of  constant 
magnitude  of  force  are  shewn  by  dotted  lines. 

Let  us  now  consider  a  body  whose  permeability  (/x)  depends 
on  the  strength  of  the  field  F.  At  two  field-strengths,  such  as 
Fa,  Ff,,  let  the  permeability  of  the  body  be  equal  to  that  of  the 


IV] 


STRUCTURE   OF   THE   CELL 


183 


medium,  and  let  the  curved  line  in  Fig.  55  represent  generally 
its  permeability  at  other  field-strengths;  and  let  the  outer  and 
inner  dotted  curves  in  Fig.  54  represent  respectively  the  loci  of 
the  field-strengths  F^  and  F^-  The  body  if  it  be  placed  in  the 
medium  within  either  branch  of  the  inner  curve,  or  outside  the 
outer  curve,  will  tend  to  move  into  the  neighbourhood  of  the 
adjacent  pole.  If  it  be  placed  in  the  region  intermediate  to  the 
two  dotted  curves,  it  will  tend  to  move  towards  regions  of  weaker 
field-strength. 

The  locus  2^6  is  therefore  a  locus  of  stable  position,  towards 
which  the  body  tends  to  move ;  the  locus  Fg,  is  a  locus  of  unstable 
position,  from  which  it  tends  to  move.     If  the  body  were  placed 


f^ 


Fig.  55. 

across  F^,  it  might  be  torn  asunder  into  two  portions,  the  split 
coinciding  with  the  locus  F^. 

Suppose  a  number  of  such  bodies  to  be  scattered  throughout 
the  medium.  Let  at  first  the  regions  F^  and  F^  be  entirely  outside 
the  space  where  the  bodies  are  situated:  and,  in  making  this 
supposition  we  may,  if  we  please,  suppose  that  the  loci  which  we 
are  calhng  F^  and  F^  are  meanwhile  situated  somewhat  farther 
from  the  axis  than  in  our  figure,  that  (for  instance)  F^  is  situated 
where  we  have  drawn  Fj,,  and  that  Fj,  is  still  further  out.  The 
bodies  then  tend  towards  the  poles ;  but  the  tendency  may  be 
very  small  if,  in  Fig.  55,  the  curve  and  its  intersecting  straight  fine 
do  not  diverge  very  far  from  one  another  beyond  Fa',    in  other 


184  ON   THE   INTERNAL  FORM  AND  [ch. 

words,  if,  when  situated  in  this  region,  the  permeabihty  of  the 
bodies  is  not  very  much  in  excess  of  that  of  the  medium. 

Let  the  poles  now  tend  to  separate  farth-er  and  farther  from 
one  another,  the  strength  of  each  pole  remaining  unaltered ;  in 
other  words,  let  the  centrosome-foci  recede  from  one  another,  as 
they  actually  do,  drawing  out  the  spindle-threads  between  them. 
The  loci  F^,  F^,,  will  close  in  to  nearer  relative  distances  from  the 
poles.  In  doing  so,  when  the  locus  F„  crosses  one  of  the  bodies, 
the  body  may  be  torn  asunder;  if  the  body  be  of  elongated  shape, 
and  be  crossed  at  more  points  than  one,  the  forces  at  work  will 
tend  to  exaggerate  its  foldings,  and  the  tendency  to  rupture  is 
greatest  when  F„  is  in  some  median  position  (Fig.  56). 

When  the  locus  F^  has  passed  entirely  over  the  body,  the  body 
tends  to  move  towards  regions  of  weaker  force ;    but  when,  in 


Fa 

Fig.  5n. 

turn,  the  locus  F^,  has  crossed  it,  then  the  body  again  moves  towards 
regions  of  stronger  force,  that  is  to  say,  towards  the  nearest  pole. 
And,  in  thus  moving  towards  the  pole,  it  will  do  so.  as  appears 
actually  to  be  the  case  in  the  dividing  cell,  along  the  course  of 
the  outer  hues  of  force,  the  so-called  "mantle-fibres"  of  the 
histologist*. 

Such  considerations  as  these  give  general  results,  easily  open 
to  modification  in  detail  by  a  change  of  any  of  the  arbitrary 
postulates  which  have  been  made  for  the  sake  of  simplicity. 
Doubtless  there  are  many  other  assumptions  which  would  more 
or  less  meet  the  case;    for  instance,  that  of  Ida  H.  Hyde  that, 

*  We  have  not  taken  account  in  the  above  paragraphs  of  the  obvious  fact  that 
the  supposed  symmetrical  field  of  force  is  distorted  by  the  presence  in  it  of  the 
more  or  less  permeable  bodies;  nor  is  it  necessary  for  us  to  do  so,  for  to  that 
distorted  field  the  above  argument  continues  to  apply,  word  for  word. 


IV]  STRUCTURE    OF   THE   CELL  185 

during  the  active  phase  of  the  chromatin  molecule  (during  which 
it  decomposes  and  sets  free  nucleic  acid)  it  carries  a  charge  opposite 
to  that  which  it  bears  during  its  resting,  or  alkaline  phase ;  and 
that  it  would  accordingly  move  towards  different  poles  under  the 
influence  of  a  current,  wandering  with  its  negative  charge  in  an 
alkahne  fluid  during  its  acid  phase  to  the  anode,  and  to  the  kathode 
during  its  alkahne  phase.  A  whole  field  of  speculation  is  opened 
up  when  we  begin  to  consider  the  cell  not  merely  as  a  polarised 
electrical  field,  but  also  as  an  electrolytic  field,  full  of  wandering 
ions.  Indeed  it  is  high  time  we  reminded  ourselves  that  we  have 
perhaps  been  deahng  too  much  with  ordinary  physical  analogies : 
and  that  our  whole  field  of  force  within  the  cell  is  of  an  order  of 
magnitude  where  these  grosser  analogies  may  fail  to  serve  us, 
and  might  even  play  us  false,  or  lead  us  astray.  But  our  sole 
object  meanwhile,  as  I  have  said  more  than  once,  is  to  demon- 
strate, by  such  illustrations  as  these,  that,  whatever  be  the  actual 
and  as  yet  unknown  modiis  operandi,  there  are  physical  conditions 
and  distributions  of  force  which  could  produce  just  such  phenomena 
of  movement  as  we  see  taking  place  within  the  living  cell. 
This,  and  no  more,  is  precisely  what  Descartes  is  said  to  have 
claimed  for  his  description  of  the  human  body  as  a  "  mechanism  *." 

The  foregoing  account  is  based  on  the  provisional  assumption 
that  the  phenomena  of  caryokinesis  are  analogous  to,  if  not  identical 
with  those  of  a  bipolar  electrical  field;  and  this  comparison,  in 
my  opinion,  offers  without  doubt  the  best  available  series  of 
analogies.  But  we  must  on  no  account  omit  to  mention  the 
fact  that  some  of  Leduc's  diffusion-experiments  offer  very  remark- 
able analogies  to  the  diagrammatic  phenomena  of  caryokinesis,  as 
shewn  in  the  annexed  figure  f.  Here  we  have  two  identical  (not 
opposite)  poles  of  osmotic  concentration,  formed  by  placing  a  drop 
of  indian  ink  in  salt  water,  and  then  on  either  side  of  this  central 
drop,  a  hypertonic  drop  of  salt  solution  more  lightly  coloured. 
On  either  side  the  pigment  of  the  central  drop  has  been  drawn 
towards  the  focus  nearest  to  it ;  but  in  the  middle  line,  the  pigment 

*  M.  Foster,  Lectures  on  the  History  of  Physiology,  1901,  p.  62. 
t  Op.  cit.  pp.  110  and  91. 


186 


ON   THE   INTERNAL  FORM  AND 


[CH. 


is  drawn  in  opposite  directions  by  equal  forces,  and  so  tends  to 
remain  undisturbed,  in  the  form  of  an  "equatorial  plate." 

Nor  should  we  omit  to  take  account  (however  briefly  and 
inadequately)  of  a  novel  and  elegant  hypothesis  put  forward  by 
A.  B.  Lamb.  This  hypothesis  makes  use  of  a  theorem  of  Bjerknes, 
to  the  effect  that  synchronously  vibrating  or  pulsating  bodies  in 
a  hquid  field  attract  or  repel  one  another  according  as  their 
oscillations  are  identical  or  opposite  in  phase.  Under  such 
circumstances,  true  currents,  or  hydrodynamic  lines  of  force,  are 
produced,  identical  in  form  with  the  hues  of  force  of  a  magnetic 
field ;  and  other  particles  floating,  though  not  necessarily  pulsating, 
in  the  hquid  field,  tend  to  be  attracted  or  repelled  by  the  pulsating 


Fig.  57.     Artificial  caryokinesis  (after  Leduc),  for  comparison 
with  Fig.  41,  p.   169. 

bodies  according  as  they  are  lighter  or  heavier  than  the  surrounding 
fluid.  Moreover  (and  this  is  the  most  remarkable  point  of  all), 
the  fines  of  force  set  up  by  the  oppositely  pulsating  bodies  are  the 
same  as  those  which  are  produced  by  opposite  magnetic  poles : 
though  in  the  former  case  repulsion,  and  in  the  latter  case  attrac- 
tion, takes  place  between  the  two  poles*. 


But  to  return  to  our  general  discussion. 

While  it  can  scarcely  be  too  often  repeated  that  our  enquiry 
is  not  directed  towards  the  solution  of  physiological  problems,  save 

*  Lamb,  A.  B.,  A  new  Explanation  of  the  Mechanism  of  Mitosis,  Journ.  Exp. 
Zool.  y,  pp.  27-33,  1908. 


IV]  STRUCTURE   OF   THE   CELL  187 

only  in  so  far  as  they  are  inseparable  from  the  problems  presented 
by  the  visible  configurations  of  form  and  structure,  and  while  we 
try,  as  far  as  possible,  to  evade  the  difficult  question  of  what 
particular  forces  are  at  work  when  the  mere  visible  forms  produced 
are  such  as  to  leave  this  an  open  question,  yet  in  this  particular 
case  we  have  been  drawn  into  the  use  of  electrical  analogies,  and 
we  are  bound  to  justify,  if  possible,  our  resort  to  this  particular 
mode  of  physical  action.  There  is  an  important  paper  by  R.  S.  Lillie, 
on  the  "Electrical  Convection  of  certain  Free  Cells  and  Nuclei*," 
which,  while  I  cannot  quote  it  in  direct  support  of  the  suggestions 
which  I  have  made,  yet  gives  just  the  evidence  we  need  in  order 
to  shew  that  electrical  forces  act  upon  the  constituents  of  the 
cell,  and  that  their  action  discriminates  between  the  two  species 
of  colloids  represented  by  the  cytoplasm  and  the  nuclear  chromatin. 
And  the  difference  is  such  that,  in  the  presence  of  an  electrical 
current,  the  cell  substance  and  the  nuclei  (including  sperm-cells) 
tend  to  migrate,  the  former  on  the  whole  with  the  positive,  the 
latter  with  the  negative  stream :  a  difference  of  electrical  potential 
being  thus  indicated  between  the  particle  and  the  surrounding 
medium,  just  as  in  the  case  of  minute  suspended  particles  of  various 
kinds  in  various  feebly  conducting  media  "j".  And  the  electrical 
difference  is  doubtless  greatest,  in  the  case  of  the  cell  constituents, 
just  at  the  period  of  mitosis :  when  the  chromatin  is  invariably 
in  its  most  deeply  staining,  most  strongly  acid,  and  therefore, 
presumably,  in  its  most  electrically  negative  phase.     In  short, 

*  Amer.  J.  of  Physiol,  via,  pp.  273-283,  1903  [vide,  supra,  p.  181);  cf.  ibid. 
XV,  pp.  46-84,  1905.     Cf.  also  Biological  Btilleti^i,  iv,  p.  175,  1903. 

f  In  Uke  manner  Hardy  has  shewn  that  colloid  particles  migrate  with  the 
negative  stream  if  the  reaction  of  the  surrounding  fluid  be  alkahne,  and  vice  versa. 
The  whole  subject  is  much  wider  than  these  brief  allusions  suggest,  and  is  essentially 
part  of  Quincke's  theory  of  Electrical  Diffusion  or  Endosmosis:  according  to 
which  the  particles  and  the  fluid  in  which  they  float  (or  the  fluid  and  the  capillary 
walls  through  which  it  flows)  each  carry  a  charge,  there  being  a  discontinuity  of 
potential  at  the  surface  of  contact,  and  hence  a  field  of  force  leading  to  powerful 
tangential  or  shearing  stresses,  communicating  to  the  particles  a  velocity  which 
varies  with  the  density  per  unit  area  of  the  surface  charge.  See  W.  B.  Hardy's 
paper  on  Coagulation  by  Electricity,  Journ.  of  Physiol.  xxn%  p.  288-304,  1899, 
also  Hardy  and  H.  W.  Harvey,  Surface  Electric  Charges  of  Living  Cells,  Proc.  R.  S. 
Lxxxiv  (B),  pp.  217-226,  1911,  and  papers  quoted  therein.  Cf.  also  E.  N.  Harvey's 
observations  on  the  convection  of  unicellular  organisms  in  an  electric  field  (Studies 
on  the  PermeabiHty  of  Cells,  Journ.  of  Exper.  Zool.  x,  pp.  508-556,  1911). 


188  ON  THE   INTERNAL   FORM  AND  [ch. 

Lillie  comes  easily  to  the  conclusion  that  "electrical  theories  of 
mitosis  are  entitled  to  more  careful  consideration  than  they  have 
hitherto  received." 

Among  other  investigations,  all  leading  towards  the  same 
general  conclusion,  namely  that  differences  of  electric  potential 
play  a  great  part  in  the  phenomenon  of  cell  division,  I  would 
mention  a  very  noteworthy  paper  by  Ida  H.  Hyde*,  in  which  the 
writer  shews  (among  other  important  observations)  that  not  only 
is  there  a  measurable  difference  of  potential  between  the  animal 
and  vegetative  poles  of  a  fertilised  egg  {Fundulus,  toad,  turtle, 
etc.),  but  that  this  difference  is  not  constant,  but  fluctuates,  or 
actually  reverses  its  direction,  periodically,  at  epochs  coinciding 
with  successive  acts  of  segmentation  or  other  important  phases 
in  the  development  of  the  eggt;  just  as  other  physical  rhythms, 
for  instance  in  the  production  of  CO2,  had  already  been  shewm 
to  do.  Hence  we  shall  be  by  no  means  surprised  to  find  that  the 
"materialised"  lines  of  force,  which  in  the  earlier  stages  form  the 
convergent  curves  of  the  spindle,  are  replaced  in  the  later  phases 
of  caryokinesis  by  divergent  curves,  indicating  that  the  two  foci, 
which  are  marked  out  within  the  field  by  the  divided  and  recon- 
stituted nuclei,  are  now  alike  in  their  polarity  (Figs.  58,  59). 

It  is  certain,  to  my  mind,  that  these  observations  of  Miss 
Hyde's,  and  of  Lillie's,  taken  together  with  those  of  many  writers 
on  the  behaviour  of  colloid  particles  generally  in  their  relation 
to  an  electrical  field,  have  a  close  bearing  upon  the  physiological 
side  of  our  problem,  the  full  discussion  of  which  lies  outside  our 
.present  field. 

The  break-up  of  the  nucleus,  already  referred  to  and  ascribed 
to  a  diminution  of  its  surface-tension,  is  accompanied  by  certain 
diffusion  phenomena  which  are  sometimes  visible  to  the  eye ;  and 
we  are  reminded  of  Lord  Kelvin's  view  that  diffusion  is  implicitly 

*  On  Differences  in  Electrical  Potential  in  Developing  Eggs,  Atner.  Journ.  of 
Physiol,  xn,  pp.  241-275,  1905.  This  paper  contains  an  excellent  summary  of 
various  physical  theories  of  the  segmentation  of  the  cell. 

t  Gray  has  recently  demonstrated  a  temporary  increase  of  electrical  con- 
ductivity in  sea-urchin  eggs  during  the  process  of  fertihsation  (The  Electrical 
Conductivity  of  fertilised  and  unfertiUsed  Eggs,  Journ.  Mar.  Biol.  Assoc,  x,  pp. 
50-59,  1913). 


IV] 


STRUCTURE   OF   THE   CELL 


189 


associated  with  surface-tension  changes,  of  which  the  first  step 
is  a  minute  puckering  of  the  surface-skin,  a  sort  of  interdigi- 
tation  with  the  surrounding  medium.  For  instance,  Schewia- 
kofE  has  observed  in  Euglypha*  that,  just  before  the  break-up 
of  the  nucleus,  a  system  of  rays  appears,  concentred  about  it, 
but  having  nothing  to  do  with  the  polar  asters :  and  during  the 
existence  of  this  striation,  the  nucleus  enlarges  very  considerably, 
evidently  by  imbibition  of  fluid  from  the  surrounding  protoplasm. 
In  short,  diffusion  is  at  work,  hand  in  hand  with,  and  as  it  were 
in  opposition   to,  the  surface-tensions  which  define  the  nucleus. 


Fig.  58.  Final  stage  in  tfte  first 
segmentation  of  the  egg  of  Cerebra- 
tulus.     (From  Prenant,  after  Coe.)| 


Fig.  59.     Diagram  of  field  of  force 
with  two  similar  poie.?. 


By  diffusion,  hand  in  hand  with  surface-tension,  the  alveoli  of 
the  nuclear  meshw^ork  are  formed,  enlarged,  and  finally  ruptured : 
diffusion  sets  up  the  movements  which  give  rise  to  the  appearance 
of  rays,  or  striae,  around  the  nucleus :  and  through  increasing 
diffusion,  and  weakening  surface-tension,  the  rounded  outline  of 
the  nucleus  finally  disappears. 

*  Schewiakoff,  Ueber  die  karyokinetische  Kerntheilung  der  Euglypha  alveolata, 
Morph.  Jahrb.  xm,  pp.  193-258.  1888  (see  p.  216). 

f  Coe,  W.  R.,  Maturation  and  Fertihzation  of  the  Egg  of  Cerebratuhis,  Zool. 
Jahrbiicher  {Anat.  Abth.),  xii,  pp.  425-476,  1899. 


190  ON   THE   INTERNAL  FORM  AND  [ch. 

As  we  study  these  manifold  phenomena,  in  the  individual  cases 
of  particular  plants  and  animals,  we  recognise  a  close  identity  of 
type,  coupled  with  almost  endless  variation  of  specific  detail ; 
and  in  particular,  the  order  of  succession  in  which  certain  of  the 
phenomena  occur  is  variable  and  irregular.  The  precise  order  of 
the  phenomena,  the  time  of  longitudinal  and  of  transverse  fission 
of  the  chromatin  thread,  of  the  break-up  of  the  nuclear  wall,  and 
so  forth,  will  depend  upon  various  minor  contingencies  and 
"interferences."  And  it  is  worthy  of  particular  note  that  these 
variations,  in  the  order  of  events  and  in  other  subordinate  details, 
while  doubtless  attributable  to  specific  physical  conditions,  would 
seem  to  be  w^ithout  any  obvious  classificatory  value  or  other 
biological  significance*. 

As  regards  the  actual  mechanical  division  of  the  cell  into  two 
halves,  we  shall  see  presently  that,  in  certain  cases,  such  as  that 
of  a  long  cyHndrical  filament,  surface-tension,  and  what  is  known 
as  the  principle  of  "minimal  area,"  go  a  long  way  to  explain  the 
mechanical  process  of  division;  and  in  all  cells  whatsoever,  the 
process  of  division  must  somehow  be  explained  as  the  result  of 
a  conflict  between  surface-tension  and  its  opposing  forces.  But 
in  such  a  case  as  our  spherical  cell,  it  is  not  very  easy  to  siee  what 
physical  cause  is  at  work  to  disturb  its  equilibrium  and  its  integrity. 

The  fact  that,  when  actual  division  of  the  cell  takes  place,  it 
does  so  at  right  angles  to  the  polar  axis  and  precisely  in  the 
direction  of  the  equatorial  plane,  would  lead  us  to  suspect  that 
the  new  surface  formed  in  the  equatorial  plane  sets  up  an  annular 
tension,  directed  inwards,  where  it  meets  the  outer  surface  layer 
of  the  cell  itself.  But  at  this  point,  the  problem  becomes  more 
complicated.  Before  we  could  hope  to  comprehend  it,  we  should 
have  not  only  to  enquire  into  the  potential  distribution  at  the 
surface  of  the  cell  in  relation  to  that  which  we  have  seen  to  exist 
in  its  interior,  but  we  should  probably  also  have  to  take  account 
of  the  differences  of  potential  which  the  material  arrangements 
along  the  lines  of  force  must  themselves  tend  to  produce.     Only 

*  Thus,  for  example,  Farmer  and  Digby  (On  Dimensions  of  Chromosomes 
considered  in  relation  to  Phylogeny,  Phil.  Trails.  (B),  ccv,  pp.  1-23,  1914)  have 
been  at  pains  to  shew,  in  confutation  of  Meek  (ibid,  ccni,  pp.  1-74,  1912),  that  the 
width  of  the  chromosomes  cannot  be  correlated  with  the  order  of  phylogeny. 


IV]  STRUCTURE   OF   THE   CELL  191 

thus  could  we  approach  a  comprehension  of  the  balance  of  forces 
which  cohesion,  friction,  capillarity  and  electrical  distribution 
combine  to  set  up. 

The  manner  in  which  we  regard  the  phenomenon  would  seem 
to  turn,  in  great  measure,  upon  whether  or  no  we  are  justified  in 
assuming  that,  in  the  hquid  surface-film  of  a  minute  spherical  cell, 
local,  and  symmetrically  locahsed,  differences  of  surface-tension 
are  likely  to  occur.  If  not,  then  changes  in  the  conformation  of 
the  cell  such  as  lead  immediately  to  its  division  must  be  ascribed 
not  to  local  changes  in  its  surface-tension,  but  rather  to  direct 
changes  in  internal  pressure,  or  to  mechanical  forces  due  to  an 
induced  surface-distribution  of  electrical  potential. 

It  has  seemed  otherwise  to  many  writers,  and  we  have  a  number 
of  theories  of  cell  division  which  are  all  based  directly  on  in- 
equalities or  asymmetry  of  surface-tension.  For  instance,  Biitschli 
suggested,  some  forty  years  ago*,  that  cell  division  is  brought 
about  by  an  increase  of  surface-tension  in  the  equatorial  region 
of  the  cell.  This  explanation,  however,  can  scarcely  hold;  for 
it  would  seem  that  such  an  increase  of  surface-tension  in  the 
equatorial  plane  would  lead  to  the  cell  becoming  flattened  out  into 
a  disc,  with  a  sharply  curved  equatorial  edge,  and  to  a  streaming 
of  material  towards  the  equator.  In  1895,  Loeb  shewed  that  the 
streaming  went  on  from  the  equator  towards  the  divided  nuclei, 
and  he  supposed  that  the  violence  of  these  streaming  movements 
brought  about  actual  division  of  the  cell :  a  hypothesis  which  was 
adopted  by  many  other  physiologists  f.  This  streaming  move- 
ment would  suggest,  as  Robertson  has  pointed  out,  a  diminution 
of  surface-tension  in  the  region  of  the  equator.  Now  Quincke  has 
shewn  that  the  formation  of  soaps  at  the  surface  of  an  oil-droplet 
results  in  a  diminution  of  the  surface-tension  of  the  latter;  and 
that  if  the  saponification  be  local,  that  part  of  the  surface  tetids  to 
spread.  By  laying  a  thread  moistened  with  a  dilute  solution  of 
caustic  alkali,  or  even  merely  smeared  with  soap,  across  a  drop 
of  oil,  Robertson  has  further  shewn  that  the  drop  at  once  divides 
into  two:    the  edges  of  the  drop,  that  is  to  say  the  ends  of  the 

*  Cf.  also  Arch.  f.  Entw.  Mech.  x,  p.  52,  1900. 

t  Cf.  Loeb,  Am.  J.  of  Physiol,  vi,  p.  432,   1902. ;    Erlanger,  Biol.  Centralbl. 
xvn,  pp.  152,  339,  1897 ;    Conklin,  Biol.  Lectures,  Woods  Holl,  p.  69,  etc.  1898-9. 


192  ON   THE   INTERNAL  FORM   AND  [ch. 

diameter  across  which  the  thread  Hes,  recede  from  the  thread, 
so  forming  a  notch  at  each  end  of  the  diameter,  while  violent 
streaming  motions  are  set  up  at  the  surface,  away  from  the  thread 
in  the  direction  of  the  two  opposite  poles.  Robertson*  suggests, 
accordingly,  that  the  division  of  the  cell  is  actually  brought  about 
by  a  lowering  of  the  equatorial  surface-tension,  and  that  this  in 
turn  is  due  to  a  chemical  action,  such  as  a  liberation  of  choUn, 
or  of  soaps  of  cholin,  through  the  splitting  of  lecithin  in  nuclear 
synthesis. 

But  purely  chemical  changes  are  not  of  necessity  the  funda- 
mental cause  of  alteration  in  the  surface-tension  of  the  egg,  for 
the  action  of  electrolytes  on  surface-tension  is  now  well  known 
and  easily  demonstrated.  So,  according  to  other  views  than 
those  with  which  we  have  been  dealing,  electrical  charges  are 
sufficient  in  themselves  to  account  for  alterations  of  surface- 
tension  ;  while  these  in  turn  account  for  that  protoplasmic 
streaming  which,  as  so  many  investigators  agree,  initiates  the 
segmentation  of  the  egg|.  A  great  part  of  our  difficulty  arises 
from  the  fact  that  in  such  a  case  as  this  the  various  phenomena 
are  so  entangled  and  apparently  concurrent  that  it  is  hard  to  say 
which  initiates  another,  and  to  which  this  or  that  secondary 
phenomenon  may  be  considered  due.  Of  recent  years  the  pheno- 
menon of  adsorptio7L  has  been  adduced  (as  we  have  already  briefly 
said)  in  order  to  account  for  many  of  the  events  and  appearances 
which  are  associated  with  the  asymmetry,  and  lead  towards  the 
division,  of  the  cell.  But  our  short  discussion  of  this  phenomenon 
may  be  reserved  for  another  chapter. 

Hov/ever,  we  are  not  directly  concerned  here  with  the 
phenomena  of  segmentation  or  cell  division  in  themselves,  except 
only  in  so  far  as  visible  changes  of  form  are  capable  of  easy  and 
obvious  correlation  with  the  play  of  force.  The  very  fact  of 
"development"  indicates  that,  while  it  lasts,  the  equihbrium  of 
the  egg  is  never  complete  J.     And  we  may  simply  conclude  the 

*  Robertson,  T.  B.,  Note  on  the  Chemical  Mechanics  of  Cell  Division,  Arclu 
/.  Entw.  Mech.  xxvn,  p.  29,  1909,  xxxv,  p.  692.  1913.  Cf.  R.  S.  Lilhe,  J.  Exp. 
Zool.  XXI,  pp.  369—402,  1916. 

t  Cf.  D'Arsonval,  Arch,  de  Physiol,  p.  460,  1889;    Ida  H.  Hyde,  op.cit.  p.  242. 

I  Cf.  Plateau's  remarks  {Statique  des  liquides,  n,  p.  154)  on  the  tendency  towarda 
equilibrium,  rather  than  actual  equilibrium,  in  many  of  his  systems  of  soap-films^ 


IV]  STRUCTURE   OF  THE   CELL  193 

matter  by  saying  that,  if  you  have  caryokinetic  figures  developing 
inside  the  cell,  that  of  itself  indicates  that  the  dynamic  system 
and  the  locahsed  forces  arising  from  it  are  in  continual  alteration ; 
and,  consequently,  changes  in  the  outward  configuration  of  the 
system  are  bound  to  take  place. 

As  regards  the  phenomena  of  fertihsation, — of  the  union  of 
the  spermatozoon  with  the  "pronucleus"  of  the  egg, — we  might 
study  these  also  in  illustration,  up  to  a  certain  point,  of  the 
polarised  forces  which  are  manifestly  at  work.  But  we  shall 
merely  take,  as  a  single  illustration,  the  paths  of  the  male  and 
female  pronuclei,  as  they  travel  to  their  ultimate  meeting  place. 

The  spermatozoon,  when  within  a  very  short  distance  of  the 
egg-cell,  is  attracted  by  it.  Of  the  nature  of  this  attractive  force 
we  have  no  certain  knowledge,  though  we  would  seem  to  have 
a  pregnant  hint  in  Loeb's  discovery  that,  in  the  neighbourhood 
of  other  substances,  such  even  as  a  fragment,  or  bead,  of  glass, 
the  spermatozoon  undergoes  a  similar  attraction.  But,  whatever 
the  force  may  be,  it  is  one  acting  normally  to  the  surface  of  the 
ovum,  and  accordingly,  after  entry,  the  sperm-nucleus  points 
straight  towards  the  centre  of  the  egg ;  from  the  fact  that  other 
spermatozoa,  subsequent  to  the  first,  fail  to  effect  an  entry,  we 
may  safely  conclude  that  an  immediate  consequence  of  the  entry 
of  the  spermatozoon  is  an  increase  in  the  surface-tension  of  the 
egg*.  Somewhere  or  other,  near  or  far  away,  within  the  egg,  hes 
its  own  nuclear  body,  the  so-called  female  pronucleus,  and  we 
find  after  a  while  that  this  has  fused  with  the  head  of  the  sperma- 
tozoon (or  male  pronucleus),  and  that  the  body  resulting  from 
their  fusion  has  come  to  occupy  the  centre  of  the  egg.  This  must 
be  due  (as  Whitman  pointed  out  long  ago)  to  a  force  of  attraction 
acting  between  the  two  bodies,  and  another  force  acting  upon 
one  or  other  or  both  in  the  direction  of  the  centre  of  the  cell. 
Did  we  know  the  magnitude  of  these  several  forces,  it  would  be 
a  very  easy  task  to  calculate  the  precise  path  which  the  two 
pronuclei  would  follow,  leading  to  conjugation  and  the  central 

*  But  under  artificial  conditions,  "polyspermy"  may  take  place,  e.g.  under 
the  action  of  dilute  poisons,  or  of  an  abnormally  high  temperature,  these  being 
all,  doubtless,  conditions  under  which  the  surface-tension  is  diminished. 

T.  G.  13 


194  ON  THE  INTERNAL  FORM  AND  [ch. 

position.  As  we  do  not  know  the  magnitude,  but  only  the  direction, 
of  these  forces  we  can  only  make  a  general  statement:  (1)  the 
paths  of  both  moving  bodies  will  lie  wholly  within  a  plane  triangle 
drawn  between  the  two  bodies  and  the  centre  of  the  cell ;  (2)  unless 
the  two  bodies  happen  to  he,  to  begin  with,  precisely  on  a  diameter 
of  the  cell,  their  paths  until  they  meet  one  another  will  be  curved 
paths,  the  convexity  of  the  curve  being  towards  the  straight  hne 
joining  the  two  bodies ;  (3)  the  two  bodies  will  meet  a  httle  before 
they  reach  the  centre ;  and,  having  met  and  fused,  will  travel 
on  to  reach  the  centre  in  a  straight  hne.  The  actual  study  and 
observation  of  the  path  followed  is  not  very  easy,  owing  to  the 
fact  that  what  we  usually  see  is  not  the  path  itself,  but  only  a 
"projection  of  the  path  upon  the  plane  of  the  microscope ;  but  the 
curved  path  is  particularly  well  seen  in  the  frog's  egg,  where  the 
path  of  the  spermatozoon  is  marked  by  a  httle  streak  of  brown 
pigment,  and  the  fact  of  the  meeting  of  the  pronuclei  before 
reaching  the  centre  has  been  repeatedly  seen  by  many  observers. 
The  problem  is  nothing  else  than  a  particular  case  of  the 
famous  problem  of  three  bodies,  which  has  so  occupied  the 
astronomers ;  and  it  is  obvious  that  the  foregoing  brief  description 
is  very  far  from  including  all  possible  cases.  Many  of  these  are 
particularly  described  in  the  works  of  Fol,  Roux,  Whitman  and 
others*. 

The  intracellular  phenomena  of  which  we  have  now  spoken 
have  assumed  immense  importance  in  biological  hterature  and 
discussion  during  the  last  forty  years ;  but  it  is  open  to  us  to  doubt 
whether  they  will  be  found  in  the  end  to  possess  more  than  a 
remote  and  secondary  biological  significance.  Most,  if  not  all  of 
them,  would  seem  to  follow  Immediately  and  inevitably  from  very 
simple  assumptions  as  to  the  physical  constitution  of  the  cell,  and 
from  an  extremely  simple  distribution  of  polarised  forces  within 
it.  We  have  already  seen  that  how  a  thing  grows,  and  what  it 
grows  into,  is  a  dynamic  and  not  a  merely  material  problem;  so 
far  as  the  material  substance  is  concerned,  it  is  so  only  by  reason 

*  Fol,  H.,  Recherches  sur  la  fecondation,  1879.  Roux,  W.,  Beitrage  zur 
Entwickelungsmechanik  des  Embryo,  Arch.  f.  Mikr.  Anat.  xix,  1887.  Whitman, 
C.  O.,  Ookinesis,  Journ.  of  Morph.  i,  1887. 


IV]  STRUCTURE   OF   THE   CELL  195 

of  the  chemical,  electrical  or  other  forces  which  are  associated 
with  it.  But  there  is  another  consideration  which  would  lead  us 
to  suspect  that  many  features  in  the  structure  and  configuration 
of  the  cell  are  of  very  secondary  biological  importance ;  and  that 
is,  the  great  variation  to  which  these  phenomena  are  subject  in 
similar  or  closely  related  organisms,  and  the  apparent  impossibility 
of  correlating  them  with  the  peculiarities  of  the  organism  as  a 
whole.  "Comparative  study  has  shewn  that  almost  every  detail 
of  the  processes  (of  mitosis)  described  above  is  subject  to  variation 
in  different  forms  of  cells*."  A  multitude  of  cells  divide  to  the 
accompaniment  of  caryokinetic  phenomena ;  but  others  do  so 
without  any  visible  caryokinesis  at  all.  Sometimes  the  polarised 
field  of  force  is  within,  sometimes  it  is  adjacent  to,  and  at  other 
times  it  Hes  remote  from  the  nucleus.  The  distribution  of  potential 
is  very  often  symmetrical  and  bipolar,  as  in  the  case  described; 
but  a  less  symmetrical  distribution  often  occurs,  with  the  result  that 
we  have,  for  a  time  at  least,  numerous  centres  of  force,  instead 
of  the  two  main  correlated  poles :  this  is  the  simple  explanation 
of  the  numerous  stellate  figures,  or  "  Strahlungen,"  which  have 
been  described  in  certain  eggs,  such  as  those  of  Cliaetofterus.  In 
one  and  the  same  species  of  worm  {Ascaris  megalocefhala),  one 
group  or  two  groups  of  chromosomes  may  be  present.  And 
remarkably  constant,  in  general,  as  the  number  of  chromosomes  in 
any  one  species  undoubtedly  is,  yet  we  must  not  forget  that,  in 
plants  and  animals  alike,  the  whole  range  of  observed  numbers  is 
but  a  small  one;  for  (as  regards  the  germ-nuclei)  few  organisms 
have  less  than  six  chromosomes,  and  fewer  still  have  more  than 
sixteen  t-  In  closely  related  animals,  such  as  various  species  of 
Copepods,  and  even  in  the  same  species  of  worm  or  insect,  the 
form  of  the  chromosomes,  and  their  arrangement  in  relation  to 
the  nuclear  spindle,  have  been  found  to  differ  in  the  various  ways 
alluded  to  above.  In  short,  there  seem  to  be  strong  grounds  for 
beheving  that  these  and  many  similar  phenomena  are  in  no  way 
specifically  related  to  the  particular  organism  in  which  they  have 

*  Wilson.  The  Cell.  p.  77. 

t  Eight  and  twelve  are  by  much  the  commonest  numbers,  six  and  sixteen 
coming  next  in  order.  If  we  may  judge  by  the  list  given  by  E.  B.  Wilson  {The 
Cell,  p.  206),  over  80  %  of  the  observed  cases  lie  between  6  and  16,  and  nearly 
60  %  between  8  and  12. 

13—2 


196  ON  THE   INTERNAL  FORM  AND  [ch. 

been  observed,  and  are  not  even  specially  and  indisputably  con- 
nected with  the  organism  as  such.  They  include  such  manifesta- 
tions of  the  physical  forces,  in  their  various  permutations  and 
combinations,  as  may  also  be  witnessed,  under  appropriate 
conditions,  in  non-living  things. 

When  we  attempt  to  separate  our  purely  morphological  or 
"purely  embryological"  studies  from  physiological  and  physical 
investigations,  we  tend  ipso  facto  to  regard  each  particular  structure 
and  configuration  as  an  attribute,  or  a  particular  "character,"  of 
this  or  that  particular  organism.  From  this  assumption  we  are 
apt  to  go  on  to  the  drawing  of  new  conclusions  or  the  framing  of 
new  theories  as  to  the  ancestral  history,  the  classificatory  position, 
the  natural  affinities  of  the  several  organisms :  in  fact,  to  apply 
our  embryological  knowledge  mainly,  and  at  times  exclusively,  to 
the  study  of  phytogeny.  When  we  find,  as  we  are  not  long  of 
finding,  that  our  phylogenetic  hypotheses,  as  drawn  from  em- 
bryology, become  complex  and  unwieldy,  we  are  nevertheless 
reluctant  to  admit  that  the  whole  method,  with  its  fundamental 
postulates,  is  at  fault.  And  yet  nothing  short  of  this  would 
seem  to  be  the  case,  in  regard  to  the  earlier  phases  at  least  of 
embryonic  development.  All  the  evidence  at  hand  goes,  as  it 
seems  to  me,  to  shew  that  embryological  data,  prior  to  and  even 
long  after  the  epoch  of  segmentation,  are  essentially  a  subject  for 
physiological  and  physical  investigation  and  have  but  the  very 
slightest  link  with  the  problems  of  systematic  or  zoological 
classification.  Comparative  embryology  has  its  own  facts  to 
classify,  and  its  own  methods  and  principles  of  classification. 
Thus  we  may  classify  eggs  according  to  the  presence  or  absence, 
the  paucity  or  abundance,  of  their  associated  food-yolk,  the 
chromosomes  according  to  their  form  and  their  number,  the 
segmentation  according  to  its  various  "types,"  radial,  bilateral, 
spiral,  and  so  forth.  But  we  have  little  right  to  expect,  and  in 
point  of  fact  we  shall  very  seldom  and  (as  it  were)  only  accidentally 
find,  that  these  embryological  categories  coincide  with  the  lines 
of  "natural"  or  "phylogenetic"  classification  which  have  been 
arrived  at  by  the  systematic  zoologist. 

The  cell,  which  Goodsir  spoke  of  as  a  "centre  of  force,"  is  in 


IV]  STKUCTUEE   OF   THE   CELL  197 

reality  a  "sphere  of  action"  of  certain  more  or  less  localised 
forces ;  and  of  these,  surface-tension  is  the  particular  force  which 
is  especially  responsible  for  giving  to  the  cell  its  outline  and  its 
morphological  individuality.  The  partially  segmented  differs  from 
the  totally  segmented  egg,  the  unicellular  Infusorian  from  the 
minute  multicellular  Turbellarian,  in  the  intensity  and  the  range  of 
those  surface-tensions  which  in  the  one  case  succeed  and  in  the 
other  fail  to  form  a  visible  separation  between  the  "cells."  Adam 
Sedgwick  used  to  call  attention  to  the  fact  that  very  often,  even 
in  eggs  that  appear  to  be  totally  segmented,  it  is  yet  impossible 
to  discover  an  actual  separation  or  cleavage,  through  and  through 
between  the  cells  which  on  the  surface  of  the  egg  are  so  clearly 
delimited ;  so  far  and  no  farther  have  the  physical  forces  effect- 
uated a  visible  "cleavage."  The  vacuolation  of  the  protoplasm  in 
Actinophrys  or  Actinosphaerium  is  due  to  localised  surface-tensions, 
quite  irrespective  of  the  multinuclear  nature  of  the  latter 
organism.  In  short,  the  boundary  walls  due  to  surface-tension 
may  be  present  or  may  be  absent  with  or  without  the  dehmi- 
nation  of  the  other  specific  fields  of  force  which  are  usually 
correlated  with  these  boundaries  and  with  the  independent 
individuality  of  the  cells.  What  we  may  safely  admit,  however, 
is  that  one  effect  of  these  circumscribed  fields  of  force  is  usually 
such  a  separation  or  segregation  of  the  protoplasmic  constituents, 
the  more  fluid  from  the  less  fluid  and  so  forth,  as  to  give  a  field 
where  surface-tension  may  do  its  work  and  bring  a  visible  boundary 
into  being.  When  the  formation  of  a  "surface"  is  once  effected, 
its  physical  condition,  or  phase,  will  be  bound  to  differ  notably 
from  that  of  the  interior  of  the  cell,  and  under  appropriate  chemical 
conditions  the  formation  of  an  actual  cell-wall,  cellulose  or  other, 
is  easily  intelligible.  To  this  subject  we  shall  return  again,  in 
another  chapter. 

From  the  moment  that  we  enter  on  a  dynamical  conception 
of  the  cell,  we  perceive  that  the  old  debates  were  in  vain  as  to 
what  visible  portions  of  the  cell  were  active  or  passive,  living  or 
non-living.  For  the  manifestations  of  force  can  only  be  due  to 
the  interaction  of  the  various  parts,  to  the  transference  of  energy 
from  one  to  another.  Certain  properties  may  be  manifested, 
certain  functions  may  be  carried  on,  by  the  protoplasm  apart 


198  ON  THE   INTERNAL  FOEM  AND  [ch. 

from  the  nucleus;  but  the  interaction  of  the  two  is  necessary, 
that  other  and  more  important  properties  or  functions  may  be 
manifested.  We  know,  for  instance,  that  portions  of  an  Infusorian 
are  incapable  of  regenerating  lost  parts  in  the  absence  of  a  nucleus, 
while  nucleated  pieces  soon  regain  the  specific  form  of  the  organism : 
and  we  are  told  that  reproduction  by  fission  cannot  be  initiated, 
though  apparently  all  its  later  steps  can  be  carried  on,  indepen- 
dently of  nuclear  action.  Nor,  as  Verworn  pointed  out,  can  the 
nucleus  possibly  be  regarded  as  the  "sole  vehicle  of  inheritance," 
since  only  in  the  conjunction  of  cell  and  nucleus  do  we  find  the 
essentials  of  cell-Hfe.  "Kern  und  Protoplasma  sind  nur  vereint 
lebensfahig,"  as  Nussbaum  said.  Indeed  we  may,  with  E.  B. 
Wilson,  go  further,  and  say  that  "the  terms  'nucleus'  and  'cell- 
body  '  should  probably  be  regarded  as  only  topographical  expres- 
sions denoting  two  difEerentiated  areas  in  a  common  structural 
basis." 

Endless  discussion  has  taken  place  regarding  the  centrosome, 
some  holding  that  it  is  a  specific  and  essential  structure,  a  per- 
manent corpuscle  derived  from  a  similar  pre-existing  corpuscle,  a 
"fertihsing  element"  in  the  spermatozoon,  a  special  "organ  of 
cell-division,"  a  material  "dynamic  centre"  of  the  cell  (as  Van 
Beneden  and  Boveri  call  it) ;  while  on  the  other  hand,  it  is  pointed 
out  that  many  cells  live  and  multiply  without  any  visible  centro- 
somes,  that  a  centrosome  inaj  disappear  and  be  created  anew, 
and  even  that  under  artificial  conditions  abnormal  chemical 
stimuli  may  lead  to  the  formation  of  new  centrosomes.  We  may 
safely  take  it  that  the  centrosome,  or  the  "attraction  sphere," 
is  essentially  a  "centre  of  force,"  and  that  this  dynamic  centre 
may  or  may  not  be  constituted  by  (but  will  be  very  apt  to  produce) 
a  concrete  and  visible  concentration  of  matter. 

It  is  far  from  correct  to  say,  as  is  often  done,  that  the  cell-wall, 
or  cell-membrane,  belongs  "  to  the  passive  products  of  protoplasm 
rather  than  to  the  living  cell  itself"  ;  or  to  say  that  in  the  animal 
cell,  the  cell-wall,  because  it  is  "shghtly  developed,"  is  relatively 
unimportant  compared  with  the  important  role  which  it  assumes 
in  plants.  On  the  contrary,  it  is  quite  certain  that,  whether 
visibly  differentiated  into  a  semi-permeable  membrane,  or  merely 
constituted  by  a  hquid  film,  the  surface  of  the  cell  is  the  seat  of 


IV]  STRUCTURE   OF  THE   CELL  199 

important  forces,  capillary  and  electrical,  which  play  an  essential 
part  in  the  dynamics  of  the  cell.  Even  in  the  thickened,  largely 
soHdified  cellulose  wall  of  the  plant-cell,  apart  from  the  mechanical 
resistances  which  it  affords,  the  osmotic  forces  developed  in  con- 
nection with  it  are  of  essential  importance. 

But  if  the  cell  acts,  after  this  fashion,  as  a  whole,  each  part 
interacting  of  necessity  with  the  rest,  the  same  is  certainly  true 
of  the  entire  multicellular  organism:  as  Schwann  said  of  old,  in 
very  precise  and  adequate  words,  "the  whole  organism  subsists 
only  by  means  of  the  reciprocal  action  of  the  single  elementary 
parts*." 

As  Wilson  says  again,  "the  physiological  autonomy  of  the 
individual  cell  falls  into  the  background... and  the  apparently 
composite  character  which  the  multicellular  organism  may  exhibit 
is  owing  to  a  secondary  distribution  of  its  energies  among  local 
centres  of  actionf.  ' 

It  is  here  that  the  homology  breaks  down  which  is  so  often 
drawn,  and  overdrawn,  between  the  unicellular  organism  and  the 
individual* cell  of  the  metazoonj. 

Whitman,  Adam  Sedgwick  §,  and  others  have  lost  no 
opportunity  of  warning  us  against  a  too  literal  acceptation 
of  the  cell-theory,  against  the  view  that  the  multicellular 
organism  is  a  colony  (or,  as  Haeckel  called  it  (in  the  case 
of  the  plant),  a  "republic")  of  independent  units  of  life||. 
As  Goethe  said  long  ago,  "Das  lebendige  ist  zwar  in  Elemente 

*  Theory  of  Cells,  p.  191. 

f  The  Cell  in  Development,  etc.  p.  59;  cf.  pp.  388,  413.. 

J  E.g.  Briicke;  Elementarorganismen,  p.  387:  "Wir  miissen  in  der  Zelle  einen 
kleinen  Thierleib  sehen,  und  diirfen  die  Analogien,  welche  zwischen  ihr  und  den 
kleinsten  Thierformen   existiren,  niemals  aus  den  Augen  lassen." 

§  Whitman,  C.  0.,  The  Inadequacy  of  the  Cell-theory,  Journ.  of  Morphol. 
vm,  pp.  639-658,  1893;  Sedgwick,  A.,  On  the  Inadequacy  of  the  Cellular  Theory 
of  Development,  Q.J. M.S.  xxxvii,  pp.  87-101,  1895,  xxxviii,  pp.  331-337,  1896. 
Cf.  Bourne,  G.  C,  A  Criticism  of  the  Cell-theory;  being  an  answer  to  Mr  Sedgwick's 
article,  etc.,  ibid,  xxxvm,  pp.   137-174,  1896. 

II  Cf.  Hertwig,  0.,  Die  Zelle  und  die  Gewebe,  1893,  p.  1;  "Die  Zellen,  in  welche 
der  Anatom  die  pflanzlichen  und  thierischen  Organismen  zerlegt,  sind  die  Trager 
der  Lebensfunktionen ;  sie  sind,  wie  Virchow  sich  ausgedriickt  hat,  die  'Lebensein- 
heiten.'  Von  diesem  Gesichtspunkt  aus  betrachtet,  erscheint  der  Gesammtlebens- 
process  eines  zusammengesetzten  Organismus  nichts  Anderes  zu  sein  als  das  hochst 
verwickelte  Resultat  der  einzelnen  Lebensprocesse  seiner  zahbeichen,  verschieden 
functionirenden  Zellen." 


200    INTERNAL  FORM  AND  STRUCTURE  OF  CELL    [ch.  iv 

zerlegt,  aber  man  kann  es  aus  diesen  nicht  wieder  zusammenstellen 
und  beleben;"  the  dictum  of  the  Cellular fathologie  being  just 
the  opposite,  "Jedes  Thier  erscheint  als  eine  Summe  vitaler 
Einheiten,  von  denen  jede  den  vollen  CharaJcter  des  Lebens  an 
sich  tragi." 

Hofmeister  and  Sachs  have  taught  us  that  in  the  plant  the 
growth  of  the  mass,  the  growth  of  the  organ,  is  the  primary  fact, 
that  "cell  formation  is  a  phenomenon  very  general  in  organic 
life,  but  still  only  of  secondary  significance."  "Comparative 
embryology"  says  Whitman,  "reminds  us  at  every  turn  that  the 
organism  dominates  cell-formation,  using  for  the  same  purpose 
one,  several,  or  many  cells,  massing  its  material  and  directing  its 
movements  and  shaping  its  organs,  as  if  cells  did  not  exist*." 
So  Rauber  declared  that,  in  the  whole  world  of  organisms,  "das 
Ganze  lief ert  die  Theile,  nicht  die  Theile  das  Ganze :  letzteres 
setzt  die  Theile  zusammen,  nicht  diese  jenes|."  And  on  the 
botanical  side  De  Bary  has  summed  up  the  matter  in  an  aphorism, 
"Die  Pflanze  bildet  Zellen,  nicht  die  Zelle  bildet  Pflanzen." 

Discussed  almost  wholly  from  the  concrete,  or  morphological 
point  of  view,  the  question  has  for  the  most  part  been  made  to  turn 
on  whether  actual  protoplasmic  continuity  can  be  demonstrated 
between  one  cell  and  another,  whether  the  organism  be  an  actual 
reticulum,  or  syncytium.  But  from  the  dynamical  point  of  view 
the  question  is  much  simpler.  We  then  deal  not  with  material 
continuity,  not  with  little  bridges  of  connecting  protoplasm,  but 
with  a  continuity  of  forces,  a  comprehensive  field  of  force,  which 
runs  through  and  through  the  entire  organism  and  is  by  no  means 
restricted  in  its  passage  to  a  protoplasmic  continuum.  And  such 
a  continuous  field  of  force,  somehow  shaping  the  whole  organism, 
independently  of  the  number,  magnitude  and  form  of  the  individual 
cells,  which  enter,  Kke  a  froth,  into  its  fabric,  seems  to  me  certainly 
and  obviously  to  exist.  As  Whitman  says,  "the  fact  that  physio- 
logical unity  is  not  broken  by  cell-boundaries  is  confirmed  in  so 
many  ways  that  it  must  be  accepted  as  one  of  the  fundamental 
truths  of  biokgyj." 

*  Journ.  of  Morph.  viii.  p.  653,  1893. 

•f  Neue  Grundlegungen  zur  Kenntniss  der  Zelle,  Morph.  Jahrb.  vin,  pp.  272, 
3]  3,  333,  1883. 

I  Journ.  of  Morph.  ii,  p.  49,  1889. 


CHAPTER   V 

THE   FORMS   OF  CELLS 

Protoplasm,  as  we  have  already  said,  is  a  fluid  or  rather  a 
semifluid  substance,  and  we  need  not  pause  here  to  attempt  to 
describe  the  particular  properties  of  the  semifluid,  colloid,  or 
jelly-like  substances  to  which  it  is  allied;  we  should  find  it  no 
easy  matter.  Nor  need  we  appeal  to  precise  theoretical  definitions 
of  fluidity,  lest  we  come  into  a  debateable  land.  It  is  in  the  most 
general  sense  that  protoplasm  is  "fluid."  As  Graham  said  (of 
colloid  matter  in  general),  "its  softness  partaJces  of  fluidity,  and 
enables  the  colloid  to  become  a  vehicle  for  liquid  diffusion,  like 
water  itself*."  When  we  can  deal  with  protoplasm  in  sufficient 
quantity  we  see  it  flow ;  particles  move  freely  through  it,  air- 
bubbles  and  hquid  droplets  shew  round  or  spherical  within  it ; 
and  we  shall  have  much  to  say  about  other  phenomena  manifested 
by  its  own  surface,  which  are  those  especially  characteristic  of 
liquids.  It  may  encompass  and  contain  solid  bodies,  and  it  may 
"secrete"  within  or  around  itself  solid  substances;  and  very 
often  in  the  complex  living  organism  these  solid  substances 
formed  by  the  living  protoplasm,  like  shell  or  nail  or  horn  or 
feather,  may  remain  when  the  protoplasm  which  formed  them 
is  dead  and  gone ;  but  the  protoplasm  itself  is  fluid  or  semifluid, 
and  accordingly  permits  of  free  (though  not  necessarily  rapid) 
diffusion  and  easy  convection  of  particles  within  itself.  This  simple 
fact  is  of  elementary  importance  in  connection  with  form,  and 
with  what  appear  at  first  sight  to  be  common  characteristics  or 
pecuharities  of  the  forms  of  living  things. 

The  older  naturalists,  in  discussing  the  differences  between 
inorganic  and  organic  bodies,  laid  stress  upon  the  fact  or  state- 
ment that  the  former  grow  by  "agglutination,"  and  the  latter  by 

*  Phil.  Trans,  cli,  p.  183,  1861;  Researches,  ed.  Angus  Smith,  1877,  p.  553. 


202  THE  FORMS   OF  CELLS  [cii. 

what  they  termed  "intussusception."  The  contrast  is  true, 
rather,  of  solid  as  compared  with  jelly-Hke  bodies  of  all  kinds, 
living  or  dead,  the  great  majority  of  which  as  it  so  happens,  but 
by  no  means  all,  are  of  organic  origin. 

A  crystal  "grows"  by  deposition  of  new  molecules,  one  by 
one  and  layer  by  layer,  superimposed  or  aggregated  upon  the 
solid  substratum  already  formed.  Each  particle  would  seem  to 
be  influenced,  practically  speaking,  only  by  the  particles  in  its 
immediate  neighbourhood,  and  to  be  in  a  state  of  freedom  and 
independence  from  the  influence,  either  direct  or  indirect,  of  its 
remoter  neighbours.  As  Lord  Kelvin  and  others  have  explained 
the  formation  and  the  resulting  forms  of  crystals,  so  we  beheve 
that  each  added  particle  takes  up  its  position  in  relation  to  its 
immediate  neighbours  already  arranged,  generally  in  the  holes  and 
corners  that  their  arrangement  leaves,  and  in  closest  contact  with 
the  greatest  number*.  And  hence  we  may  repeat  or  imitate  this 
process  of  arrangement,  with  great  or  apparently  even  with 
precise  accuracy  (in  the  case  of  the  simpler  crystalline  systems), 
by  piling  up  spherical  pills  or  grains  of  shot.  Li  so  doing,  we  must 
have  regard  to  the  fact  that  each  particle  must  drop  into  the 
place  where  it  can  go  most  easily,  or  where  no  easier  place  offers. 
In  more  technical  language,  each  particle  is  free  to  take  up,  and 
does  take  up,  its  position  of  least  potential  energy  relative  to  those 
already  deposited;  in  other  words,  for  each  particle  motion  is 
induced  until  the  energy  of  the  system  is  so  distributed  that  no 
tendency  or  resultant  force  remains  to  move  it  more.  The 
application  of  this  principle  has  been  shewn  to  lead  to  the  produc- 
tion of  planes -f  (in  all  cases  where  by  the  limitation  of  material, 
surfaces  must  occur) ;  and  where  we  have  planes,  straight  edges 
and  solid  angles  must  obviously  also  occur ;   and,  if  equilibrium  is 

*  Cf.  Kelvin,  On  the  Molecular  Tactics  of  a  Crystal,  The  Boyle  Lecture,  Oxford, 
1893,  Baltimore  Lectures,  1904,  pp.  612-642.  Here  Kelvin  was  mainly  following 
Bravais's  (tod  Frankenheim's)  theory  of  "space-lattices,"  but  he  had  been  largely 
anticipated  by  the  crystallographers.  For  an  account  of  the  development  of  the 
subject  in  modern  crystallography,  by  Sohncke,  von  Fedorow,  Schonfiiess,  Barlow 
and  others,  see  Tutton's  Crystallography,  chap,  ix,  pp.  118-134,  1911. 

f  In  a  homogeneous  crystalline  arrangement,  symmetry  compels  a  locus  of  one 
property  to  be  a  plane  or  set  of  planes;  the  locus  in  this  case  being  that  of  least 
surface  potential  energy. 


V]  OF  CRYSTALLINE   AND   COLLOID   BODIES        203 

to  follow,  must  occur  symmetrically.  Our  piling  up  of  shot,  or 
manufacture  of  mimic  crystals,  gives  us  visible  demonstration 
that  the  result  is  actually  to  obtain,  as  in  the  natural  crystal, 
plane  surfaces  and  sharp  angles,  symmetrically  disposed. 

But  the  living  cell  grows  in  a  totally  dift'erent  way,  very  much 
as  a  piece  of  glue  swells  up  in  water,  by  "imbibition,"  or  by  inter- 
penetration  into  and  throughout  its  entire  substance.  The  semi- 
fluid colloid  mass  takes  up  water,  partly  to  combine  chemically 
with  its  individual  molecules*,  partly  by  physical  diffusion  into 
the  interstices  between  these  molecules,  and  partly,  as  it  would 
seem,  in  other  ways ;  so  that  the  entire  phenomenon  is  a  very 
complex  and  even  an  obscure  one.  But,  so  far  as  we  are  con- 
cerned, the  net  result  is  a  very  simple  one.  For  the  equilibrium  or 
tendency  to  equilibrium  of  fluid  pressure  in  all  parts  of  its  interior 
while  the  process  of  imbibition  is  going  on,  the  constant  rearrange- 
ment of  its  fluid  mass,  the  contrast  in  short  with  the  crystalline 
method  of  growth  where  each  particle  comes  to  rest  to  move 
(relatively  to  the  whole)  no  more,  lead  the  mass  of  jelly  to  swell 
up,  very  much  as  a  bladder  into  which  we  blow  air,  and  so,  by 
a  graded  and  harmonious  distribution  of  forces,  to  assume  every- 
where a  rounded  and  more  or  less  bubble-like  external  formf. 
So,  when  the  same  school  of  older  naturahsts  called  attention  to 
a  new  distinction  or  contrast  of  form  between  the  organic  and 
inorganic  objects,  in  that  the  contours  of  the  former  tended  to 
roundness  and  curvature,  and  those  of  the  latter  to  be  bounded 
by  straight  lines,  planes  and  sharp  angles,  we  see  that  this  contrast 
was  not  a  new  and  different  one,  but  only  another  aspect  of 
their  former  statement,  and  an  immediate  consequence  of  the 
difference  between  the  processes  of  agglutination  and  intussus- 
ception. 

This  common  and  general  contrast  between  the  form  of  the 
crystal  on  the  one  hand,  and  of  the'  colloid  or  of  the  organism  on 
the  other,  must  by  no  means  be  pressed  too  far.     For  Lehmann, 

*  This  is  what  Graham  called  the  water  of  gelatination,  on  the  analogy  of  xvater 
of  crystallisation ;  Chem.  and  Phys.  Researches,  p.  597. 

f  Here,  in  a  non-crystalline  or  random  arrangement  of  particles,  symmetry 
ensures  that  the  potential  energy  shall  be  the  same  per  unit  area  of  all  surfaces; 
and  it  follows  from  geometrical  considerations  that  the  total  surface  energy  will 
be  least  if  the  surface  be  spherical. 


204  THE  FORMS   OF  CELLS  [ch. 

in  his  great  work  on  so-called  Fluid  Crystals*,  to  which  we  shall 
afterwards  return,  has  shewn  how,  under  certain  circumstances, 
surface-tension  phenomena  may  coexist  with  crystallisation,  and 
produce  a  form  of  minimal  potential  which  is  a  resultant  of  both : 
the  fact  being  that  the  bonds  maintaining  the  crystalline  arrange- 
ment are  now  so  much  looser  than  in  the  solid  condition  that  the 
tendency  to  least  total  surface-area  is  capable  of  being  satisfied. 
Thus  the  phenomenon  of  "liquid  crystallisation"  does  not  destroy 
the  distinction  between  crystalline  and  colloidal  forms,  but  gives 
added  unity  and  continuity  to  the  whole  series  of  phenomena  f. 
Lehmann  has  also  demonstrated  phenomena  within  the  crystal, 
known  for  instance  as  transcrystallisation,  which  shew  us  that  we 
must  not  speak  unguardedly  of  the  growth  of  crystals  as  limited 
to  deposition  upon  a  surface,  and  Biitschli  has  already  pointed  out 
the  possible  great  importance  to  the  biologist  of  the  various 
phenomena  which  Lehmann  has  described t. 

So  far  then,  as  growth  goes  on,  unafltected  by  pressure  or  other 
external  force,  the  fluidity  of  protoplasm,  its  mobility  internal 
and  external,  and  the  manner  in  which  particles  move  with 
comparative  freedom  from  place  to  place  within,  all  manifestly 
tend  to  the  production  of  swelling,  rounded  surfaces,  and  to  their 
great  predominance  over  plane  surfaces  in  the  contour  of  the 
organism.  These  rounded  contours  will  tend  to  be  preserved,  for 
a  while,  in  the  case  of  naked  protoplasm  by  its  viscosity,  and  in 
.  the  presence  of  a  cell- wall  by  its  very  lack  of  fluidity.  In  a  general 
way,  the  presence  of  curved  boundary  surfaces  will  be  especially 
obvious  in  the  unicellular  organisms,  and  still  more  generally  in 
the  external  forms  of  all  organisms ;  and  wherever  mutual  pressure 
between  adjacent  cells,  or  other  adjacent  parts,  has  not  come  into 
play  to  flatten  the  rounded  surfaces  into  planes. 

But  the  rounded  contours  that  are  assumed  and  exhibited  by 

*  Lehmann,  0.,  Flussige  Krysfalle,  soivie  Plasticitdt  von  Krystallen  im  allge- 
meinen,  etc.,  264  pp.  39  plL,  Leipsig,  1904.  For  a  semi-popular,  illustrated  account, 
see  Tutton's  Crystals  (Int.  Soi.  Series),  1911. 

t  As  Graham  said  of  an  allied  phenomenon  (the  so-called  blood-crystals  of 
Funke),  it  "illustrates  the  maxim  that  in  nature  there  are  no  abrupt  transitions, 
and  that  distinctions  of  class  are  never  absolute." 

X  Cf.  Przibram,  H.,  Kristall-analogien  zur  Entwickelungsmechanik  der  Organ- 
ismen,  Arch.  f.  Entw.  Mech.  xxii,p.  207, 1906  (with  copious  bibliography);  Lehmann, 
Scheinbar  lebende  Kristalle  und  MyeUnformen,  ibid.  xxAa,  p.  483,  1908. 


V]  OF  SURFACE   TENSION  '  205 

a  piece  of  hard  glue,  when  we  throw  it  into  water  and  see  it  expand 
as  it  sucks  the  water  up,  are  not  nearly  so  regular  or  so  beautiful 
as  are  those  which  appear  when  we  blow  a  bubble,  or  form  a 
drop,  or  pour  water  into  a  more  or  less  elastic  bag.  For  these 
curving  contours  depend  upon  the  properties  of  the  bag  itself, 
of  the  film  or  membrane  that  contains  the  mobile  gas,  or  that 
contains  or  bounds  the  mobile  liquid  mass.  And  hereby,  in  the 
case  of  the  fluid  or  semifluid  mass,  we  are  introduced  to  the 
subject  of  surface  tension :  of  which  indeed  we  have  spoken  in 
the  preceding  chapter,  but  which  we  must  now  examine  with 
greater  care. 

Among  the  forces  which  determine  the  forms  of  cells,  whether 
they  be  solitary  or  arranged  in  contact  with  one  another,  this 
force  of  surface-tension  is  certainly  of  great,  and  is  probably  of 
paramount  importance.  But  while  we  shall  try  to  separate  out 
the  phenomena  which  are  directly  due  to  it,  we  must  not  forget 
that,  in  each  particular  case,  the  actual  conformation  which  we 
study  may  be,  and  usually  is,  the  more  or  less  complex  resultant 
of  surface  tension  acting  together  with  gravity,  mechanical 
pressure,  osmosis,  or  other  physical  forces. 

Surface  tension  is  that  force  by  which  we  explain  the  form  of 
a  drop  or  of  a  bubble,  of  the  surfaces  external  and  internal  of 
a  "froth"  or  collocation  of  bubbles,  and  of  many  other  things  of 
like  nature  and  in  like  circumstances*.  It  is  a  property  of  liquids 
(in  the  sense  at  least  with  which  our  subject  is  concerned),  and  it 
is  manifested  at  or  very  near  the  surface,  where  the  liquid  comes 
into  contact  with  another  liquid,  a  solid  or  a  gas.  We  note  here 
that  the  term  surface  is  to  be  interpreted  in  a  wide  sense ;  for 
wherever  Ave  have  solid  particles  imbedded  in  a  fluid,  wherever 
we  have  a  non-homogeneous  fluid  or  semi-fluid  such  as  a  particle 

*  The  idea  of  a  "surface-tension"  in  liquids  was  first  enunciated  by  Segner, 
De  figuris  superficierum  fluidarum,  in  Comment.  Soc.  Boy.  Gottiiigen,  1751,  p.  301. 
Hooke,  in  the  Micrographia  (1665,  Obs.  viii,  etc.),  had  called  attention  to  the 
globular  or  spherical  form  of  the  httle  morsels  of  steel  struck  off  by  a  flint,  and  had 
shewn  how  to  make  a  powder  of  such  spherical  grains,  by  heating  fine  filings  to 
melting  point.  "This  Phaenomenon"  he  said  "proceeds  from  a  propriety  which 
belongs  to  all  kinds  of  fluid  Bodies  more  or  less,  and  is  caused  by  the  Incongruity 
of  the  Ambient  and  included  Fluid,  which  so.  acts  and  modulates  each  other,  that 
they  acquire,  as  neer  as  is  possible,  a  spherical  or  globular  form...." 


206  THE  FORMS   OF   CELLS  [ch. 

of  protoplasm,  wherever  we  have  the  presence  of  "impurities,"  as 
in  a  mass  of  molten  metal,  there  we  have  always  to  bear  in  mind 
the  existence  of  "surfaces"  and  of  surface  tensions,  not  only 
on  the  exterior  of  the  mass  but  also  throughout  its  interstices, 
wherever  like  meets  unhke. 

Surface  tension  is  due  to  molecular  force,  to  force  that  is  to 
say  arising  from  the  action  of  one  molecule  upon  another,  and  it 
is  accordingly  exerted  throughout  a  small  thickness  of  material, 
comparable  to  the  range  of  the  molecular  forces.  We  imagine 
that  within  the  interior  of  the  liquid  mass  such  molecular  inter- 
actions negative  one  another:  but  that  at  and  near  the  free 
surface,  within  a  layer  or  film  approximately  equal  to  the  range 
of  the  molecular  force,  there  must  be  a  lack  of  such  equilibrium 
and  consequently  a  manifestation  of  force. 

The  action  of  the  molecular  forces  has  been  variously  explained. 
But  one  simple  explanation  (or  mode  of  statement)  is  that  the 
molecules  of  the  surface  layer  (whose  thickness  is  definite  and 
constant)  are  being  constantly  attracted  into  the  interior  by  those 
which  are  more  deeply  situated,  and  that  consequently,  as 
molecules  keep  quitting  the  surface  for  the  interior,  the  bulk  of 
the  latter  increases  while  the  surface  diminishes ;  and  the  process 
continues  till  the  surface  itself  has  become  a  minimum,  the  surface- 
shrinkage  exhibiting  itself  as  a  surface-tension.  This  is  a  sufficient 
description  of  t|;ie  phenomenon  in  cases  where  a  portion  of  liquid 
is  subject  to  no  other  than  its  oivn  molecular  forces,  and  (since  the 
sphere  has,  of  all  solids,  the  smallest  surface  for  a  given  volume) 
it  accounts  for  the  spherical  form  of  the  raindrop,  of  the  grain 
of  shot,  or  of  the  living  cell  in  many  simple  organisms.  It  accounts 
also,  as  we  shall  presently  see,  for  a  great  number  of  much  more 
complicated  forms,  manifested  under  less  simple  conditions. 

Let  us  here  briefly  note  that  surface  tension  is,  in  itself,  a 
comparatively  small  force,  and  easily  measurable:  for  instance 
that  of  water  is  equivalent  to  but  a  few  grains  per  linear  inch, 
or  a  few  grammes  per  metre.  But  this  small  tension,  when  it 
exists  in  a  curved  surface  of  very  great  curvature,  gives  rise  to  a 
very  great  pressure  directed  towards  the  centre  of  curvature.  We 
can  easily  calculate  this  pressure,  and  so  satisfy  ourselves  that, 
when  the  radius  of  curvature  is    of   molecular    dimensions,    the 


V]  OF   SURFACE   ENERGY  207 

pressure  is  of  the  magnitude  of  thousands  of  atmospheres, — a  con- 
clusion which  is  supported  by  other  physical  considerations. 

The  contraction  of  a  liquid  surface  and  other  phenomena  of 
surface  tension  involve  the  doing  of  work,  and  the  power  to  do 
work  is  what  we  call  energy.  It  is  obvious,  in  such  a  simple  case 
as  we  have  just  considered,  that  the  whole  energy  of  the  system 
is  diffused  throughout  its  molecules ;  but  of  this  whole  stock  of 
energy  it  is  only  that  part  which  comes  into  play  at  or  very  near 
to  the  surface  which  normally  manifests  itself  in  work,  and  hence 
we  may  speak  (though  the  term  is  open  to  some  objections)  of 
a  specific  surface  energy.  The  consideration  of  surface  energy, 
and  of  the  manner  in  which  its  amount  is  increased  and  multiphed 
by  the  multiplication  of  surfaces  due  to  the  subdivision  of  the 
organism  into  cells,  is  of  the  highest  importance  to  the  physiologist ; 
and  even  the  morphologist  cannot  wholly  pass  it  by,  if  he  desires 
to  study  the  form  of  the  cell  in  its  relation  to  the  phenomena  of 
surface  tension  or  "capillarity."  The  case  has  been  set  forth  with 
the  utmost  possible  lucidity  by  Tait  and  by  Clerk  Maxwell,  on 
whose  teaching  the  following  paragraphs  are  based :  they  having 
based  their  teaching  upon  that  of  Gauss, — who  rested  on  Laplace. 

Let  E  be  the  whole  potential  energy  of  a  mass  M  of  liquid; 
let  Cq  be  the  energy  per  unit  mass  of  the  interior  liquid  (we  may 
call  it  the  internal  energy) ;  and  let  e  be  the  energy  per  unit  mass 
for  a  layer  of  the  skin,  of  surface  S,  of  thickness  t,  and  density 
p  {e  being  what  we  call  the  surface  energy).  It  is  obvious  that  the 
total  energy  consists  of  the  internal  flus  the  surface  energy,  and 
that  the  former  is  distributed  through  the  whole  mass,  minus  its 
surface  layers.     That  is  to  say,  in  mathematical  language, 

E={M-S  .  ^tp)  e^  +  S  .  lltpe. 

But  this  is  equivalent  to  writing : 

=  Mco  +  S  .i:tp{e  -  Co) ; 

and  this  is  as  much  as  to  say  that  the  total  energy  of  the  system 
may  be  taken  to  consist  of  two  portions,  one  uniform  throughout 
the  whole  mass,  and  another,  which  is  proportional  on  the  one  hand 
to  the  amount  of  surface,  and  on  the  other  hand  is  proportional 
to  the  difference  between  e  and  e^,  that  is  to  say  to  the  difference 
between  the  unit  values  of  the  internal  and  the  surface  energy. 


208  THE  FORMS   OF   CELLS  [ch. 

It  was  Gauss  who  first  shewed  after  this  fashion  how,  from 
the  mutual  attractions  between  all  the  particles,  we  are  led  to  an 
expression  which  is  what  we  now  call  the  fotential  energy  of  the 
system;  and  we  know,  as  a  fundamental  theorem  of  dynamics, 
that  the  potential  energy  of  the  system  tends  to  a  minimum,  and 
in  that  minimum  finds,  as  a  matter  of  course,  its  stable  equilibrium. 

We  see  in  our  last  equation  that  the  term  Me^  is  irreducible, 
save  by  a  reduction  of  the  mass  itself.  But  the  other  term  may 
be  diminished  (1)  by  a  reduction  in  the  area  of  surface,  S,  or 
(2)  by  a  tendency  towards  equality  of  e  and  Cq,  that  is  to  say  by 
a  diminution  of  the  specific  surface  energy,  e. 

These  then  are  the  two  methods  by  which  the  energy  of  the 
system  will  manifest  itself  in  work.  The  one,  which  is  much  the 
more  important  for  our  purposes,  leads  always  to  a  diminution  of 
surface,  to  the  so-called  "principle  of  minimal  areas" ;  the  other, 
which  leads  to  the  lowering  (under  certain  circumstances)  of 
surface  tension,  is  the  basis  of  the  theory  of  Adsorption,  to  which 
we  shall  have  some  occasion  to  refer  as  the  modus  operandi  in  the 
development  of  a  cell- wall,  and  in  a  variety  of  other  histological 
phenomena.  In  the  technical  phraseology  of  the  day,  the 
"capacity  factor"  is  involved  in  the  one  case,  and  the  "intensity 
factor"  in  the  other. 

Inasmuch  as  we  are  concerned  with  the  form  of  the  cell  it  is 
the  former  which  becomes  our  main  postulate :  telling  us  that 
the  energy  equations  of  the  surface  of  a  cell,  or  of  the  free  surfaces 
of  cells  partly  in  contact,  or  of  the  partition-surfaces  of  cells  in 
contact  with  one  another  or  with  an  adjacent  solid,  all  indicate 
a  minimum  of  potential  energy  in  the  system,  by  which  the  system 
is  brought,  ipso  facto,  into  equilibrium.  And  we  shall  not  fail  to 
observe,  with  something  more  than  mere  historical  interest  and 
curiosity,  how  deeply  and  intrinsically  there  enter  into  this  whole 
class  of  problems  the  "principle  of  least  action"  of  Maupertuis, 
the  "lineae  curvae  maximi  minimive  froprietate  gaiidentes"  of 
Euler,  by  which  principles  these  old  natural  philosophers  explained 
correctly  a  multitude  of  phenomena,  and  drew  the  lines  whereon 
the  foundations  of  great  part  of  modern  physics  are  well  and 
truly  laid. 


V]  THE   MEANING  OF  SYMMETRY  209 

111  all  cases  where  the  principle  of  maxima  and  minima  comes 
into  play,  as  it  conspicuously  does  in  the  systems  of  liquid  films 
which  are  governed  by  the  laws  of  surface-tension,  the  figures  and 
conformations  produced  are  characterised  by  obvious  and  remark- 
able symmetry.  Such  symmetry  is  in  a  high  degree  characteristic 
of  organic  forms,  and  is  rarely  absent  in  living  things, — save  in  such 
cases  as  amoeba,  where  the  equilibrium  on  which  symmetry  depends 
is  likewise  lacking.  And  if  we  ask  what  physical  equilibrium  has 
to  do  with  formal  symmetry  and  regularity,  the  reason  is  not  far 
to  seek ;  nor  can  it  be  putbetter  than  in  the  following  words  of 
Mach's*.  "In  every  symmetrical  system  every  deformation  that 
tends  to  destroy  the  symmetry  is  complemented  by  an  equal  and 
opposite  deformation  that  tends  to  restore  it.  In  each  deformation 
positive  and  negative  work  is  done.  One  condition,  therefore, 
though  not  an  absolutely  sufficient  one,  that  a  maximum  or 
minimum  of  work  corresponds  to  the  form  of  equilibrium,  is  thus 
supplied  by  symmetry.  Regularity  is  successive  symmetry. 
There  is  no  reason,  therefore,  to  be  astonished  that  the  forms  of 
equilibrium  are  often  symmetrical  and  regular." 

As  we  proceed  in  our  enquiry,  and  especially  when  we  approach 
the  subject  of  tissues,  or  agglomerations  of  cells,  we  shall  have 
from  time  to  time  to  call  in  the  help  of  elementary  mathematics. 
But  already,  with  very  little  mathematical  help,  we  find  ourselves 
in  a  position  to  deal  with  some  simple  examples  of  organic  forms. 

When  we  melt  a  stick  of  sealing-wax  in  the  flame,  surface 
tension  (which  was  ineffectively  present  in  the  solid  but  finds  play 
in  the  now  fluid  mass),  rounds  off  its  sharp  edges  into  curves,  so 
striving  towards  a  surface  of  minimal  area ;  and  in  like  manner, 
by  melting  the  tip  of  a  thin  rod  of  glass,  Leeuwenhoek  made  the 
little  spherical  beads  which  served  him  for  a  microscope  "j".  When 
any  drop  of  protoplasm,  either  over  all  its  surface  or  at  some  free 
end,  as  at  the  extremity  of  the  pseudopodium  of  an  amoeba,  is 

*  Science  of  Mechanics,  1902,  p.  395 ;  see  also  Mach's  article  Ueber  die  physika- 
lische  Bedeutung  der  Gesetze  der  Symnietrie,  Lotos,  xxi,  pp.  139-147,  1871. 

t  Similarly,  Sir  David  Brewster  and  others  made  powerful  lenses  by  simply 
dropping  small  drops  of  Canada  balsam,  castor  oil,  or  other  strongly  refractive 
liquids,  on  to  a  glass  plate:  On  New  Philosophical  Instruments  (Description  of  a 
new  Fluid  Microscope),  Edinburgh,  1813,  p.  413. 

T.  G.  14 


210  '  THE  FORMS   OF  CELLS  [ch. 

seen  likewise  to  "round  itself  off,"  that  is  not  an  effect  of  "vital 
contractility,"  but  (as  Hofmeister  shewed  so  long  ago  as  1867) 
a  simple  consequence  of  surface  tension ;  and  almost  immediately 
afterwards  Engelmann*  argued  on  the  same  lines,  that  the  forces 
which  cause  the  contraction  of  protoplasm  in  general  may  "be 
just  the  same  as  those  which  tend  to  make  every  non-spherical 
drop  of  fluid  become  spherical ! "  We  are  not  concerned  here  with 
the  many  theories  and  speculations  which  would  connect  the 
phenomena  of  surface  tension  with  contractility,  muscular  move- 
ment or  other  special  'physiological  functions,  but  we  find  ample 
room  to  trace  the  operation  of  the  same  cause  in  producing,  under 
conditions  of  rest  and  equilibrium,  certain  definite  and  inevitable 
forms  of  surface. 

It  is  however  of  great  importance  to  observe  that  the  living 
cell  is  one  of  those  cases  where  the  phenomena  of  surface  tension 
are  by  no  means  limited  to  the  ovter  surface ;  for  within  the 
heterogeneous  substance  of  the  cell,  between  the  protoplasm  and 
its  nuclear  and  other  contents,  and  in  the  alveolar  network  of  the 
cytoplasm  itself  (so  far  as  that  "alveolar  structure"  is  actually 
present  in  life),  we  have  a  multitude  of  interior  surfaces ;  and, 
especially  among  plants,  we  may  have  a  large,  inner  surface  of 
"interfacial"  contact,  where  the  protoplasm  contains  cavities 
or  "vacuoles"  filled  with  a  different  and  more  fluid  material,  the 
"cell-sap."  Here  we  have  a  great  field  for  the  development  of 
surface  tension  phenomena :  and  so  long  ago  as  1865,  Nageli  and 
Schwendener  shewed  that  the  streaming  currents  of  plant  cells 
might  be  very  plausibly  explained  by  this  phenomenon.  Even 
ten  years  earlier,  Weber  had  remarked  upon  the  resemblance 
between  these  protoplasmic  streamings  and  the  streamings  to  be 
observed  in  certain  inanimate  drops,  for  which  no  cause  but 
surface  tension  could  be  assigned  f. 

The  case  of  amoeba,  though  it  is  an  elementary  case,  is  at  the 

same  time  a  complicated  one.     While  it  remains  "amoeboid,"  it 

is  never  at  rest  or  in  equilibrium;   it  is  always  moving,  from  one 

to  another  of  its  protean  changes  of  configuration ;    its  surface 

tension  is  constantly  varying  from  point  to  point.     Where  the 

*  Beitrage  z.  Physiologie  d.  Protoplasma,  Pfluger''s  Archiv,  n,  p.  307,  1869. 
t  Poggend.   Annalen,   xciv,   pp.   447-459,    1855.     Cf.   Strethill  Wright,  Phil. 
Mag.  Feb.   1860. 


V]  THE   FORM   OF  AMOEBA  211 

surface  tension  is  greater,  that  portion  of  the  surface  will  contract 
into  spherical  or  spheroidal  forms ;  where  it  is  less  the  surface 
will  correspondingly  extend.  While  generally  speaking  the  surface 
energy  has  a  minimal  value,  it  is  not  necessarily  constant.  It  may 
be  diminished  by  a  rise  of  temperature ;  it  may  be  altered  by 
contact  with  adjacent  substances*,  by  the  transport  of  constituent 
materials  from  the  interior  to  the  surface,  or  again  by  actual 
chemical  and  fermentative  change.  Within  the  cell,  the  surface 
energies  developed  about  its  heterogeneous  contents  will  constantly 
vary  as  these  contents  are  affected  by  chemical  metabolism.  As 
the  colloid  materials  are  broken  down  and  as  the  particles  in 
suspension  are  diminished  in  size  the  "free  surface  energy' 
will  be  increased,  but  the  osmotic  energy  will  be  diminished  f. 
Thus  arise  the  various  fluctuations  of  surface  tension  and  the 
various  phenomena  of  amoeboid  form  and  motion,  which  Biitschli 
and  others  have  reproduced  or  imitated  by  means  of  the  fine 
emulsions  which  constitute  their  "artificial  amoebae."  A  multi- 
tude of  experiments  shew  how  extraordinarily  delicate  is  the 
adjustment  of  the  surface  tension  forces,  and  how  sensitive  they 
are  to  the  least  change  of  temperature  or  chemical  state.  Thus, 
on  a  plate  which  we  have  warmed  at  one  side,  a  drop  of  alcohol 
runs  towards  the  warm  area,  a  drop  of  oil  away  from  it ;  and  a 
drop  of  water  on  the  glass  plate  exhibits  lively  movements  when 

*  Haycraft  and  Carlier  pointed  out  {Proc.  E.S.E.  xv,  pp.  220-224,  1888)  that 
the  amoeboid  movements  of  a  white  blood-corpuscle  are  only  manifested  when  the 
corpuscle  is  in  contact  with  some  soUd  substance:  while  floating  freely  in  the 
plasma  or  serum  of  the  blood,  these  corpuscles  are  spherical,  that  is  to  say  they 
are  at  rest  and  in  equihbrium.  The  same  fact  has  recently  been  recorded  anew 
by  Ledingham  (On  Phagocytosis  from  an  adsorptive  point  of  view,  Journ.  of  Hygiene, 
xu,  p.  324,  1912).  On  the  emission  of  pseudopodia  as  brought  about  by  changes 
in  surface  tension,  see  also  (int.  al.)  Jensen,  Ueber  den  Geotropismus  niederer 
Organismen,  Pflilger's  Archiv,  Liii,  1893.  Jensen  remarks  that  in  OrbitoUtes,  the 
pseudopodia  issuing  through  the  pores  of  the  shell  first  float  freely,  then  as  they 
grow  longer  bend  over  tiU  they  touch  the  ground,  whereupon  they  begin  to  display 
amoeboid  and  streaming  motions.  Verworn  indicates  (Allg.  Physiol.  189.5,  p.  429), 
and  Davenport  says  {Experim.  Morphology,  ii,  p.  376)  that  "this  persistent  cUnging 
to  the  substratum  is  a  '  thigmotropic '  reaction,  and  one  which  belongs  clearly  to 
the  category  of  'response.'"  (Cf.  Piitter,  Thigmotaxis  bei  Protisten,  A.  f.  Physiol. 
1900,  Suppl.  p.  247.)  But  it  is  not  clear  to  my  mind  that  to  account  for  this 
simple  phenomenon  we  need  invoke  other  factors  than  gravity  and  surface-action. 

t  Cf.  Pauli,  Allgemeine  physikalische  Chemie  d.  Zellen  u.  Gewebe,  in  Asher-Spiro's 
Ergebnisse  der  Physiologic,  1912;    Przibram,  Vitalitdt,  1913,  p.  6. 

14—2 


212  THE   FORMS   OF   CELLS  [ch. 

we  bring  into  its  neighbourhood  a  heated  wire,  or  a  glass  rod 
dipped  in  ether.  When  we  find  that  a  plasmodium  of  Aethalium, 
for  instance,  creeps  towards  a  damp  spot,  or  towards  a  warm  spot, 
or  towards  substances  that  happen  to  be  nutritious,  and  again 
creeps  away  from  solutions  of  sugar  or  of  salt,  we  seem  to  be 
dealing  with  phenomena  every  one  of  which  can  be  paralleled  by 
ordinary  phenomena  of  surface  tension*.  Even  the  soap-bubble 
itself  is  imperfectly  in  equilibrium,  for  the  reason  that  its  film, 
like  the  protoplasm  of  amoeba  or  Aethalium,  is  an  excessively 
heterogeneous  substance.  Its  surface  tensions  vary  from  point 
to  point,  and  chemical  changes  and  changes  of  temperature 
increase  and  magnify  the  variation.  The  whole  surface  of  the 
bubble  is  in  constant  movement  as  the  concentrated  portions  of 
the  soapy  fluid  make  their  way  outwards  from  the  deeper  layers ; 
it  thins  and  it  thickens,  its  colours  change,  currents  are  set  up  in 
it,  and  little  bubbles  glide  over  it;  it  continues  in  this  state  of 
constant  movement,  as  its  parts  strive  one  with  another  in  all 
their  interactions  towards  equilibrium f. 

In  the  case  of  the  naked  protoplasmic  cell,  as  the  amoeboid 
phase  is  emphatically  a  phase  of  freedom  and  activity,  of  chemical 
and  physiological  change,  so,  on  the  other  hand,  is  the  spherical 
form  indicative  of  a  phase  of  rest  or  comparative  inactivity.  In 
the  one  phase  we  see  unequal  surface  tensions  manifested  in  the 
creeping  movements  of  the  amoeboid  body,  in  the  rounding  o£E 
of  the  ends  of  the  pseudopodia,  in  the  flowing  out  of  its  substance 
over  a  particle  of  "food,"  and  in  the  current-motions  in  the  interior 
of  its  mass ;  till  finally,  in  the  other  phase,  when  internal  homo- 
geneity and  equilibrium   have  been  attained  and   the   potential 

*  The  surface-tension  theory  of  protoplasmic  movement  has  been  denied  by 
many.  Cf.  (e.g.),  Jennings,  H.  S.,  Contributions  to  the  Study  of  the  Behaviour 
of  the  Lower  Organisms,  Carnegie  Inst.  1904,  pp.  130-230;  Delhnger,  O.  P., 
Locomotion  of  Amoebae,  etc.  Journ.  Exp.  Zool.  iii,  pp.  337-357,  1906;  also  various 
papers  by  Max  Heidenhain,  in  Anatom.  Hefte  (Merkel  und  Bonnet),  etc. 

"]"  These  various  movements  of  a  liquid  surface,  and  other  still  more  striking 
movements  such  as  those  of  a  piece  of  camphor  floating  on  water,  were  at  one  time 
ascribed  by  certain  physicists  to  a  peculiar  force,  sui  generis,  the  force  e'pipolique 
of  Dutrochet :  imtil  van  der  Mensbrugghe  shewed  that  differences  of  surface  tension 
were  enough  to  account  for  this  whole  series  of  phenomena  (Sur  la  tension  super- 
ficielle  des  hquides  consideree  au  point  de  vue  de  certains  mouvements  observes 
a  leur  surface,  Mem.  Cour.  Acad,  de  Belgique,  xxxiv,  1869;   cf.  Plateau,  p.  283). 


V]  THE   FORM   OF  AMOEBA  213 

energy  of  the  system  is  for  the  time  being  at  a  minimum,  the 
cell  assumes  a  rounded  or  spherical  form,  passing  into  a  state 
of  "rest,"  and  (for  a  reason  which  we  shall  presently  see) 
becoming  at  the  same  time  "encysted." 

In  a  budding  yeast-cell  (Fig.  60),  we  see  a  more  definite  and 
restricted  change  of  surface  tension.     When  a   "bud"   appears, 
whether  with  or  without  actual  growth  by  osmosis 
or  otherwise  of  the  mass,  it  does  so  because  at  a 
certain  part  of  the  cell-surface  the  surface  tension 
has   more   or  less   suddenly  diminished,   and  the 
area  of  that  portion  expands  accordingly ;   but  in 
turn  the  surface  tension  of  the  expanded  area  will 
make  itself  felt,  and  the  bud  will  be  rounded  of?         p-     gQ^ 
into  a  more  or  less  spherical  form. 

The  yeast-cell  with  its  bud  is  a  simple  example  of  a  principle 
which  we  shall  find  to  be  very  important.  Our  whole  treatment 
of  cell-form  in  relation  to  surface-tension  depends  on  the  fact 
(which  Errera  was  the  first  to  point  out,  or  to  give  clear  expression 
to)  that  the  incipient  cell-wall  retains  with  but  little  impairment 
the  properties  of  a  liquid  film  *,  and  that  the  growing  cell,  in  spite 
of  the  membrane  by  which  it  has  already  begun  to  be  surrounded, 
behaves  very  much  like  a  fluid  drop.  But  even  the  ordinary 
yeast-cell  shows,  by  its  ovoid  and  non-spherical  form,  that  it  has 
acquired  its  shape  under  the  influence  of  some  force  other  than 
that  uniform  and  symmetrical  surface-tension  which  would  be 
productive  of  a  sphere ;  and  this  or  any  other  asymmetrical  form, 
once  acquired,  may  be  retained  by  virtue  of  the  solidification  and 
consequent  rigidity  of  the  membranous  wall  of  ^he  cell.  Unless 
such  rigidity  ensue,  it  is  plain  that  such  a  conformation  as  that  of 
the  cell  with  its  attached  bud  could  not  be  long  retained,  amidst 
the  constantly  varying  conditions,  as  a  figure  of  even  partial 
equilibrium.  But  as  a  matter  of  fact,  the  cell  in  this  case  is  not 
in  equilibrium  at  all;  it  is  in  process  of  budding,  and  is  slowly 
altering  its  shape  by  rounding  off  the  bud.  It  is  plain  that  over 
its  surface  the  surface-energies  are  unequally  distributed,  owing 
to  some  heterogeneity  of  the  substance;  and  to  this  matter  we 
shall    afterwards    return.     In    like    manner    the    developing    egg 

*  Cf.  infra,  p.  306. 


214  THE   FORMS   OF   CELLS  [ch. 

through  all  its  successive  phases  of  form  is  never  in  complete 
equilibrium ;  but  is  merely  responding  to  constantly  changing 
conditions,  by  phases  of  partial,  transitory,  unstable  and  con- 
ditional equilibrium. 

It  is  obvious  that  there  are  innumerable  solitary  plant-cells, 
and  unicellular  organisms  in  general,  which,  like  the  yeast-cell,  do 
not  correspond  to  any  of  the  simple  forms  that  may  be  generated 
under  the  influence  of  simple  and  homogeneous  surface-tension ; 
and  in  many  cases  these  forms,  which  we  should  expect  to  be 
unstable  and  transitory,  have  become  fixed  and  stable  by  reason 
of  the  comparatively  sudden  or  rapid  solidification  of  the  envelope. 
This  is  the  case,  for  instance,  in  many  of  the  more  complicated  forms 
of  diatoms  or  of  desmids,  where  we  are  dealing,  in  a  less  striking 
but  even  more  curious  way  than  in  the  budding  yeast-cell,  not 
with  one  simple  act  of  formation,  but  with  a  complicated  result 
of  successive  stages  of  localised  growth,  interrupted  by  phases  of 
partial  consolidation.  The  original  cell  has  acquired  or  assumed 
a  certain  form,  and  then,  under  altering  conditions  and  new 
distributions  of  energy,  has  thickened  here  or  weakened  there, 
and  has  grown  out  or  tended  (as  it  were)  to  branch,  at  particular 
points.  We  can  often,  or  indeed  generally,  trace  in  each  particular 
stage  of  growth  or  at  each  particular  temporary  growing  point, 
the  laws  of  surface  tension  manifesting  themselves  in  what  is 
for  the  time  being  a  fluid  surface ;  nay  more,  even  in  the  adult 
and  completed  structure^  we  have  little  difficulty  in  tracing  and 
recognising  (for  instance  in  the  outline  of  such  a  desmid  as  Euas- 
trum)  the  rounded  lobes  that  have  successively  grown  or  flowed 
out  from  the  original  rounded  and  flattened  cell.  What  we  see  in 
a  many  chambered  foraminifer,  such  as  Globigerina  or  Rotalia,  is 
just  the  same  thing,  save  that  it  is  carried  out  in  greater  complete- 
ness and  perfection.  The  little  organism  as  a  whole  is  not  a  figure 
of  equilibrium  or  of  minimal  area ;  but  each  new  bud  or  separate 
chamber  is  such  a  figure,  conditioned  by  the  forces  of  surface 
tension,  and  superposed  upon  the  complex  aggregate  of  similar 
bubbles  after  these  latter  have  become  consolidated  one  by  one 
into  a  rigid  system. 

Let  us  now  make  some  enquiry  regarding  the  various  forms 


V]  OF  LIQUID   FILMS  215 

wliich,  under  the  influence  of  surface  tension,  a  surface  can  possibly 
assume.  In  doing  so,  we  are  obviously  limited  to  conditions 
under  which  other  forces  are  relatively  unimportant,  that  is  to 
say  where  the  "surface  energy"  is  a  considerable  fraction  of 
the  whole  energy  of  the  system ;  and  this  in  general  will  be 
the  case  when  we  are  dealing  with  portions  of  liquid  so  small 
that  their  dimensions  come  within  what  we  have  called  the 
molecular  range,  or,  more  generally,  in  which  the  "specific 
surface"  is  large*:  in  other  words  it  will  be  small  or  minute 
organisms,  or  the  small  cellular  elements  of  larger  organisms, 
whose  forms  will  be  governed  by  surface-tension ;  while  the 
general  forms  of  the  larger  organisms  will  be  due  to  other  and 
non-molecular  forces.  For  instance,  a  large  surface  of  water  sets 
itself  level  because  here  gravity  is  predominant;  but  the  surface 
of  water  in  a  narrow  tube  is  manifestly  curved,  for  the  reason 
that  we  are  here  dealing  with  particles  which  are  mutually  within 
the  range  of  each  other's  molecular  forces.  The  same  is  the  case 
with  the  cell-surfaces  and  cell-partitions  which  we  are  presently 
to  study,  and  the  effect  of  gravity  will  be  especially  counteracted 
and  concealed  when,  as  in  the  case  of  protoplasm  in  a  watery 
fluid,  the  object  is  immersed  in  a  liquid  of  nearly  its  own  specific 
gravity. 

We  have  already  learned,  as  a  fundamental  law  of  surface- 
tension  phenomena,  that  a  liquid  film  in  equilibrium  assumes  a 
form  which  gives  it  a  minimal  area  under  the  conditions  to  which 
it  is  subject.  And  these  conditions  include  (1)  the  form  of  the 
boundary,  if  such  exist,  and  (2)  the  pressure,  if  any,  to  which  the 
film  is  subject;  which  pressure  is  closely  related  to  the  volume, 
of  air  or  of  liquid,  which  the  film  (if  it  be  a  closed  one)  may  have 
to  contain.  In  the  simplest  of  cases,  when  we  take  up  a  soap- 
film  on  a  plane  wire  ring,  the  film  is  exposed  to  equal  atmospheric 
pressure  on  both  sides,  and  it  obviously  has  its  minimal  area  in 
the  form  of  a  plane.  So  long  as  our  wire  ring  lies  in  one  plane 
(however  irregular  in  outline),  the  film  stretched  across  it  will 
still  be  in  a  plane ;  but  if  we  bend  the  ring  so  that  it  lies  no  longer 
in  a  plane,  then  our  film  will  become  curved  into  a  surface  which 
may  be  extremely  complicated,  but  is  still  the  smallest  possible 

*  Cf.  p.  32. 


216  THE  FORMS   OF  CELLS  [ch. 

surface  which  can  be  drawn  continuously  across  the  uneven 
boundary. 

The  question  of  pressure  involves  not  only  external  pressures 
acting  on  the  film,  but  also  that  which  the  film  itself  is  capable 
of  exerting.  For  we  have  seen  that  the  film  is  always  contracting 
to  its  smallest  limits ;  and  when  the  film  is  curved,  this  obviously 
leads  to  a  pressure  directed  inwards, — perpendicular,  that  is  to 
say,  to  the  surface  of  the  film.  In  the  case  of  the  soap-bubble, 
the  uniform  contraction  of  whose  surface  has  led  to  its  spherical 
form,  this  pressure  is  balanced  by  the  pressure  of  the  air  within ; 
and  if  an  outlet  be  given  for  this  air,  then  the  bubble  contracts 
with  perceptible  force  until  it  stretches  across  the  mouth  of  the 
tube,  for  instance  the  mouth  of  the  pipe  through  which  we  have 
blown  the  bubble.  A  precisely  similar  pressure,  directed  inwards, 
is  exercised  by  the  surface  layer  of  a  drop  of  water  or  a  globule 
of  mercury,  or  by  the  surface  pellicle  on  a  portion  or  "drop"  of 
protoplasm.  Only  we  must  always  remember  that  in  the  soap- 
bubble,  or  the  bubble  which  a  glass-blower  blows,  there  is  a  twofold 
pressure  as  compared  with  that  which  the  surface-film  exercises 
on  the  drop  of  liquid  of  which  it  is  a  part ;  for  the  bubble  consists 
(unless  it  be  so  thin  as  to  consist  of  a  mere  layer  of  molecules*) 
of  a  liquid  layer,  with  a  free  surface  within  and  another  without, 
and  each  of  these  two  surfaces  exercises  its  own  independent  and 
coequal  tension,  and  corresponding  pressure  |. 

If  we  stretch  a  tape  upon  a  flat  table,  whatever  be  the  tension 
of  the  tape  it  obviously  exercises  no  pressure  upon  the  table 
below.  But  if  we  stretch  it  over  a  curved  surface,  a  cylinder  for 
instance,  it  does  exercise  a  downward  pressure ;  and  the  more 
curved  the  surface  the  greater  is  this  pressure,  that  is  to  say  the 
greater  is  this  share  of  the  entire  force  of  tension  which  is  resolved 
in  the  downward  direction.  In  mathematical  language,  the 
pressure  (/j)  varies  directly  as  the  tension  {T),  and  inversely  as 
the  radius  of  curvature  {R) :  that  is  to  say,  p  =  T/R,  per  unit  of 
surface. 

*  Or,  more  strictly  speaking,  unless  its  thickness  be  less  than  twice  the  range 
of  the  molecular  forces. 

I  It  follows  that  the  tension,  depending  only  on  the  surface-conditions,  is 
independent  of  the  thickness  of  the  film. 


V]  OF  LIQUID   FILMS  217 

If  instead  of  a  cylinder,  which  is  curved  only  in  one  direction, 
we  take  a  case  where  there  are  curvatures  in  two  dimensions  (as 
for  instance  a  sphere),  then  the  effects  of  these  riiust  be  simply 
added  to  one  another,  and  the  resulting  pressure  j)  is  equal  to 
T/R  +  T/R'  or  p^TiljR+  l/R') *. 

And  if  in  addition  to  the  pressure  p,  which  is  due  to  surface 
tension,  we  have  to  take  into  account  other  pressures,  7?',  j)",  etc., 
which  are  due  to  gravity  or  other  forces,  then  we  may  say  that 
the  total  'pressure,  P  =  p'  +  p"  +  T  {l/R  +  l/R').  While  in  some 
cases,  for  instance  in  speaking  of  the  shape  of  a  bird's  egg,  we 
shall  have  to  take  account  of  these  extraneous  pressures,  in  the 
present  part  of  our  subject  we  shall  for  the  most  part  be  able  to 
neglect  them. 

Our  equation  is  an  equation  of  equilibrium.  The  resistance 
to  compression, — the  pressure  outwards, — of  our  fluid  mass,  is  a 
constant  quantity  (P) ;  the  pressure  inwards,  T  {l/R  +  1/-R'),  is 
also  constant;  and  if  (unlike  the  case  of  the  mobile  amoeba)  the 
surface  be  homogeneous,  so  that  T  is  everywhere  equal,  it  follows 
that  throughout  the  whole  surface  l/R  +  l/R'  -=  C  (a  constant). 

Now  equilibrium  is  attained  after  the  surface  contraction  has 
done  its  utmost,  that  is  to  say  when  it  has  reduced  the  surface 
to  the  smallest  possible  area ;  and  so  we  arrive,  from  the  physical 
side,  at  the  conclusion  that  a  surface  such  that  l/R  +  l/R'  =  C, 
in  other  words  a  surface  which  has  the  same  mean  curvature  at 
all  points,  is  equivalent  to  a  surface  of  minimal  area :  and  to  the 
same  conclusion  we  may  also  arrive  through  purely  analytical 
mathematics.  It  is  obvious  that  the  plane  and  the  sphere  are  two 
examples  of  such  surfaces,  for  in  both  cases  the  radius  of  curvature 
is  everywhere  constant,  being  equal  to  infinity  in  the  case  of  the 
plane,  and  to  some  definite  magnitude  in  the  case  of  the  sphere. 

From  the  fact  that  we  may  extend  a  soap-film  across  a  ring  of 
wire  however  fantastically  the  latter  may  be  bent,  we  realise  that 
there  is  no  limit  to  the  number  of  surfaces  of  minimal  area  which 
may  be  constructed  or  may  be  imagined ;  and  while  some  of  these 
are  very  complicated  indeed,  some,  for  instance  a  spiral  helicoid 
screw,  are  relatively  very  simple.     But  if  we  limit  ourselves  to 

*  This  simple  but  immensely  important  formula  is  due  to  Laplace  {Mecanique 
Celeste,  Bk  x.  suppl.  Theorie  de  Vaction  capillaire,  1806). 


218  THE   FORMS   OF   CELLS  [oh. 

surfaces  of  revolution  (that  is  to  say,  to  surfaces  symmetrical  about 
an  axis),  we  find,  as  Plateau  was  the  first  to  shew,  that  those  which 
meet  the  case  are  very  few  in  number.  They  are  six  in  all, 
namely  the  plane,  the  sphere,  the  cylinder,  the  catenoid,  the 
unduloid,  and  a  curious  surface  which  Plateau  called  the  nodoid. 

These  several  surfaces  are  all  closely  related,  and  the  passage 
from  one  to  another  is  generally  easy.  Their  mathematical  inter- 
relation is  expressed  by  the  fact  (first  shewn  by  Delaunay*,  in  1841) 
that  the  plane  curves  by  whose  rotation  they  are  generated  are 
themselves  generated  as  "roulettes"  of  the  conic  sections. 

Let  us  imagine  a  straight  line  upon  which  a  circle,  an  elHpse 
or  other  conic  section  rolls ;  the  focus  of  the  conic  section  will 
describe  a  line  in  some  relation  to  the  fixed  axis,  and  this  line 
(or  roulette),  rotating  around  the  axis,  will  describe  in  space  one  or 
other  of  the  six  surfaces  of  revolution  with  which  we  are  dealing. 

If  we  imagine  an  ellipse  so  to  roll  over  a  line,  either  of  its  foci 
will  describe  a  sinuous  or  wavy  line  (Fig.   61  b)  at  a  distance 


alternately  maximal  and  minimal  from  the  axis ;  and  this  wavy 
line,  by  rotation  about  the  axis,  becomes  the  meridional  line  of 
the  surface  which  we  call  the  unduloid.  The  more  unequal  the 
two  axes  are  of  our  elhpse,  the  more  pronounced  will  be  the 
sinuosity  of  the  described  roulette.  If  the  two  axes  be  equal, 
then  our  elhpse  becomes  a  circle,  and  the  path  described  by  its 
rolhng  centre  is  a  straight  line  parallel  to  the  axis  (A) ;  and 
obviously  the  solid  of  revolution  generated  therefrom  will  be  a 
cylinder.  If  one  axis  of  our  ellipse  vanish,  while  the  other  remain 
of  finite  length,  then  the  ellipse  is  reduced  to  a  straight  line,  and 
its  roulette  will  appear  as  a  succession  of  semicircles  touching  one 
another  upon  the  axis  (C) ;  the  solid  of  revolution  will  be  a  series  of 
equal  spheres.  If  as  before  one  axis  of  the  elhpse  vanish,  but  the 
other  be  infinitely  long,  then  the  curve  described  by  the  rotation 

*  Sur  la  surface  de  revolution  dont  la  courbure  moyenne  est  constante,  Journ. 
de  M.  LiouvilU,  vi,  p.  309,  1841. 


v]  OF  MINIMAL  SURFACES  219 

of  this  latter  will  be  a  circle  of  infinite  radius,  i.e.  a  straight  line 
infinitely  distant  from  the  axis  ;  and  the  surface  of  rotation  is  now 
a  'plane.  If  we  imagine  one  focus  of  our  ellipse  to  remain  at  a 
given  distance  from  the  axis,  but  the  other  to  become  infinitely 
remote,  that  is  tantamount  to  saying  that  the  ellipse  becomes 
transformed  into  a  parabola ;  and  by  the  rolling  of  this  curve 
along  the  axis  there  is  described  a  catenary  (D),  whose  solid  of 
revolution  is  the  catenoid. 

Lastly,  but  this  is  a  little  more  difl&cult  to  imagine,  we  have 
the  case  of  the  hyperbola. 

We  cannot  well  imagine  the  hyperbola  rolling  upon  a  fixed 
straight  line  so  that  its  focus  shall  describe  a  continuous  curve. 
But  let  us  suppose  that  the  fixed  line  is,  to  begin  with,  asymptotic 
to  one  branch  of  the  hyperbola,  and  that  the  rolling  proceed 
until  the  line  is  now  asymptotic  to  the  other  branch,  that  is  to 
say  touching  it  at  an  infinite  distance ;  there  will  then  be  mathe- 
matical continuity  if  we  recommence  rolling  with  this  second 
branch,  and  so  in  turn  with  the  other,  when  each  has  run  its 
course.  We  shall  see,  on  reflection,  that  the  line  traced  by  one 
and  the  same  focus  will  be  an  "  elastic  curve  "  describing  a  suc- 
cession of  kinks  or  knots  (E),  and  the  solid  of  revolution  described 
by  this  meridional  line  about  the  axis  is  the  so-called  nodoid. 

The  physical  transition  of  one  of  these  surfaces  into  another 
can  be  experimentally  illustrated  by  means  of  soap-bubbles,  or 
better  still,  after  the  method  of  Plateau,  by  means -of  a  large 
globule  of  oil,  supported  when  necessary  by  wire  rings,  within  a 
fluid  of  specific  gravity  equal  to  its  own. 

To  prepare  a  mixture  of  alcohol  and  water  of  a  density  precisely 
equal  to  that  of  the  oil-globule  is  a  troublesome  matter,  and  a 
method  devised  by  Mr  C.  R.  Darling  is  a  great  improvement  on 
Plateau's  *.  Mr  Darling  uses  the  oily  liquid  orthotoluidene,  which 
does  not  mix  with  water,  has  a  beautiful  and  conspicuous  red 
colour,  and  has  precisely  the  same  density  as  water  when  both 
are  kept  at  a  temperature  of  24°  C.  We  have  therefore  only  to 
run  the  liquid  into  water  at  this  temperature  in  order  to  produce 
beautifully  spherical  drops  of  any  required  size :    and  by  adding 

*  See  Liquid  Drops  and  Globules,  1914,  p.  11.  Robert  Boyle  used  turpentine 
in  much  the  same  way.     For  other  methods  see  Plateau,  op.  cit.  p.  154. 


220  THE   FORMS   OF   CELLS  [ch. 

a  little  salt  to  the  lower  layers  of  water,  the  drop  may  be  made 
to  float  or  rest  upon  the  denser  liquid. 

We  have  already  seen  that  the  soap-bubble,  spherical  to  begin 
with,  is  transformed  into  a  plane  when  we  relieve  its  internal 
pressure  and  let  the  film  shrink  back  upon  the  orifice  of  the  pipe. 
If  we  blow  a  small  bubble  and  then  catch  it  up  on  a  second  pipe, 
so  that  it  stretches  between,  we  may  gradually  draw  the  two  pipes 
apart,  with  the  result  that  the  spheroidal  surface  will  be  gradually 
flattened  in  a  longitudinal  direction,  and  the  bubble  w^U  be  trans- 
formed into  a  cylinder.  But  if  we  draw  the  pipes  yet  farther 
apart,  the  cylinder  will  narrow  in  the  middle  into  a  sort  of  hour- 
glass form,  the  increasing  curvature  of  its  transverse  section  being 
balanced  by  a  gradually  increasing  negative  curvature  in  the 
longitudinal  section.  The  cylinder  has,  in  turn,  been  converted 
into  an  unduloid.  When  we  hold  a  portion  of  a  soft  glass  tube  in 
the  flame,  and  "draw  it  out,"  we  are  in  the  same  identical  fashion 


Fig.  62. 

converting  a  cylinder  into  an  unduloid  (Fig.  62  a)  ;  when  on  the 
other  hand  we  stop  the  end  and  blow,  we  again  convert  the 
cylinder  into  an  unduloid  (b),  but  into  one  which  is  now  positively, 
while  the  former  was  negatively  curved.  The  two  figures  are 
essentially  the  same,  save  that  the  two  halves  of  the  one  are 
reversed  in  the  other. 

That  spheres,  cylinders  and  unduloids  are  of  the  commonest 
occurrence  among  the  forms  of  small  unicellular  organism,  or  of 
individual  cells  in  the  simpler  aggregates,  and  that  in  the  processes 
of  growth,  reproduction  and  development  transitions  are  frequent 
from  one  of  these  forms  to  another,  is  obvious  to  the  naturalist, 
and  we  shall  deal  presently  with  a  few  illustrations  of  these 
phenomena. 

But  before  we  go  further  in  this  enquiry,  it  will  be  necessary 
to  consider,  to  some  small  extent  at  least,  the  curvatures  of  the 
six  different  surfaces,  that  is  to  say,  to  determine  what  modification 


V]  OF  MINIMAL   SURFACES  221 

is  required,  in  each  case,  of  the  general  equation  which  appHes 
to  them  all.  We  shall  find  that  with  this  question  is  closely 
connected  the  question  of  the  pressures  exercised  by,  or  im- 
pinging on  the  film,  and  also  the  very  important  question  of 
the  limitations  which,  from  the  nature  of  the  case,  exist  to 
prevent  the  extension  of  certain  of  the  figures  beyond  certain 
bounds.  The  whole  subject  is  mathematical,  and  we  shall  only 
deal   with  it  in   the   most   elementary  way. 

We  have  seen  that,  in  our  general  formula,  the  expression 
1/R  +  1/R'  =  C,  a  constant;  and  that  this  is,  in  all  cases,  the 
condition  of  our  surface  being  one  of  minimal  area.  In  other 
words,  it  is  always  true  for  one  and  all  of  the  six  surfaces  which 
we  have  to  consider.  But  the  constant  C  may  have  any  value, 
positive,  negative,  or  nil. 

In  the  case  of  the  plane,  where  R  and  R'  are  both  infinite,  it 
is  obvious  that  1/R  +  1/R'  =  0.  The  expression  therefore  vanishes, 
and  our  dynamical  equation  of  equilibrium  becomes  P  =  p.  In 
short,  we  can  only  have  a  plane  film,  or  we  shall  only  find  a  plane 
surface  in  our  cell,  when  on  either  side  thereof  we  have  equal 
pressures  or  no  pressure  at  all.  A  simple  case  is  the  plane  partition 
between  two  equal  and  similar  cells,  as  in  a  filament  of  spirogyra. 

In  the  case  of  the  sphere,  the  radii  are  all  equal,  R  ^  R' ; 
they  are  also  positive,  and  T  {1/R  +  1/R'),  or  2T/R,  is  a  positive 
quantity,  involving  a  positive  pressure  P,  on  the  other  side  of  the 
equation. 

In  the  cylinder,  one  radius  of  curvature  has  the  finite  and 
positive  value  R ;  but  the  other  is  infinite.  Our  formula  becomes 
T/R,  to  which  corresponds  a  positive  pressure  P,  supplied  by  the 
surface-tension  as  in  the  case  of  the  sphere,  but  evidently  of  just 
half  the  magnitude  developed  in  the  latter  case  for  a  given  value 
of  the  radius  R. 

The  catenoid  has  the  remarkable  property  that  its  curvature  in 
one  direction  is  precisely  equal  and  opposite  to  its  curvature  in 
the  other,  this  ^property  holding  good  for  all  points  of  the  surface. 
That  is  to  say,  R  =  —  R' ;   and  the  expression  becomes 

{1/R+  1/R')  ^  {1/R- 1/R)  =  0; 
in  other  words,  the  surface,  as  in  the  case  of  the  plane,  has  no 


222  THE  FOEMS   OF  CELLS  [ch. 

curvature,  and  exercises  no  pressure.  There  are  no  other  surfaces, 
save  these  two,  which  share  this  remarkable  property ;  and  it 
follows,  as  a  simple  corollary,  that  we  may  expect  at  times  to  have 
the  catenoid  and  the  plane  coexisting,  as  parts  of  one  and  the 
same  boundary  system;  just  as,  in  a  cylindrical  drop  or  cell,  the 
cylinder  is  capped  by  portions  of  spheres,  such  that  the  cylindrical 
and  spherical  portions  of  the  wall  exert  equal  positive  pressures. 
In  the  unduloid,  unlike  the  four  surfaces  which  we  have  just 
been  considering,  it  is  obvious  that  the  curvatures  change  from 
one  point  to  another.  At  the  middle  of  one  of  the  swollen 
portions,  or  "beads,"  the  two  curvatures  are  both  positive;  the 
expression  {IjR  +  1/-R')  is  therefore  positive,  and  it  is  also  finite. 
The  film,  accordingly,  exercises  a  positive  tension  inwards,  which 
must  be  compensated  by  a  finite  and  positive  outward  pressure 
P.  At  the  middle  of  one  of  the  narrow  necks,  between  two 
adjacent  beads,  there  is  obviously,  in  the  transverse  direction, 
a  much  stronger  curvature  than  in  the  former  case,  and  the  curva- 
ture which  balances  it  is  now  a  negative  one.  But  the  sum  of  the 
two  must  remain  positive,  as  well  as  constant ;  and  we  therefore 
see  that  the  convex  or  positive  curvature  must  always  be  greater 
than  the  concave  or  negative  curvature  at  the  same  point.  This 
is  plainly  the  case  in  our  figure  of  the  unduloid. 

The  nodoid  is,  like  the  unduloid,  a  continuous  curve  which 
keeps  altering  its  curvature  as  it  alters  its  distance  from  the  axis ; 
but  in  this  case  the  resultant  pressure  inwards  is  negative  instead 
of  positive.  But  this  curve  is  a  complicated  one,  and  a  full 
discussion  of  it  would  carry  us  beyond  our  scope. 

In  one  of  Plateau's  experiments,  a  bubble  of  oil  (protected  from 
gravity  by  the  specific  gravity  of  the  surrounding  fluid  being 
identical  with  its  own)  is  balanced  between  two 
annuli.  It  may  then  be  brought  to  assume  the  form 
of  Fig.  63,  that  is  to  say  the  form  of  a  cylinder  with 
spherical  ends ;  and  there  is  then  everywhere,  owing 
to  the  convexity  of  the  surface  filjn,  a  pressure 
inwards  upon  the  fluid  contents  of  the  bubble.  If 
the  surrounding  liquid  be  ever  so  little  heavier  or 
lighter  than  that  which  constitutes  the  drop,  then 
the    conditions    of    equihbrium    will    be    accordingly 


V] 


OF  FIGURES   OF  EQUILIBRIUM 


223 


modified,  and  the  cylindrical  drop  will  assume  the  form  of  an 

miduloid  (Fig.  64  a,  b),  with  its  dilated  portion  below  or  above, 

as  the  case  may  be ;  and  our  cyUnder 

may   also,   of    course,   be    converted 

into  an  unduloid  either  by  elongating 

it  further,  or  by  abstracting  a  portion 

of    its    oil,   until    at    length    rupture 

ensues   and  the  cylinder    breaks   up 

into  two  new  spherical  drops.     In  all 

cases    alike,   the   unduloid,   hke    the 

original  cylinder,  will  be  capped  by  ^^' 

spherical  ends,  which  are  the  sign,  and  the  consequence,  of  the 

positive  pressure  produced  by  the  curved  walls  of  the  unduloid. 

But  if  our  initial  cylinder,  instead  of  being  tall,   be  a   fiat  or 

dumpy  one  (with  certain  definite  relations  of  height  to  breadth), 

then  new  phenomena   may  be  exhibited.      For  now,  if  a  little 

oil  be  cautiously   withdrawn  from  the  mass  by  help  of  a  small 

syringe,    the  cylinder  may  be   made   to   flatten   down   so   that 

its  upper  and  lower  surfaces  become  plane ;    which  is  of  itself 

an   indication   that   the   pressure   inwards  is  now  nil.      But  at 

the  very  moment  when  the  upper  and  lower  surfaces   become 

plane,   it   will   be  found   that  the  sides   curve   inwards,   in   the 

fashion  shewn  in  Fig.  65  b.     This  figure  is  a  catenoid,  which,  as 


B 


Fig.  65. 


we  have  already  seen,  is,  hke  the  plane  itself,  a  surface  exercising 
no  pressure,  and  which  therefore  may  coexist  with  the  plane  as 
part  of  one  and  the  same  system.  We  may  continue  to  withdraw 
more  oil  from  our  bubble,  drop  by  drop,  and  now  the  upper  and 
lower  surfaces  dimple  down  into  concave  portions  of  spheres,  as 
the  result  of  the  negative  internal  pressure ;  and  thereupon  the 
peripheral  catenoid  surface  alters  its  form  (perhaps,  on  this  small 
scale,  imperceptibly),  and  becomes  a  portion  of  a  nodoid  (Fig.  65  a). 


224  THE  FORMS   OF   CELLS  [ch. 

It    represents,    in    fact,    that    portion    of    the    nodoid,    which 

in  Fig.  66  hes  between  such  points  as  o,  p.     While  it  is  easy  to 

draw  the  outhne,  or  meridional 

section,    of    the    nodoid    (as   in 

Fig.  66),  it  is  obvious  that  the 

sohd  of  revolution  to  be  derived 

from  it,  can  never  be  reahsed  in 

its  entirety :  for  one  part  of  the 

solid  figure  would   cut,  or   en- 
Fie.  66.  . 

tangle  with,  another.     All  that 

we  can  ever  do,  accordingly,  is  to  realise  isolated  portions  of  the 
nodoid. 

If,  in  a  sequel  to  the  preceding  experiment  of  Plateau's,  we 
use  solid  discs  instead  of  annuli,  so  as  to  enable  us  to  exert  direct 
mechanical  pressure  upon  our  globule  of  oil,  we  again  begin  by 
adjusting  the  pressure  of  these  discs  so  that  the  oil  assumes  the 
form  of  a  cyhnder:  our  discs,  that  is  to  say,  are  adjusted  to 
exercise  a  mechanical  pressure  equal  to  what  in  the  former  case 
was  supphed  by  the  surface-tension  of  the  spherical  caps  or  ends 
of  the  bubble.  If  we  now  increase  the  pressure  shghtly,  the 
peripheral  walls  will  become  convexly  curved,  exercising  a  pre- 
cisely corresponding  pressure.  Under  these  circumstances  the 
form  assumed  by  the  sides  of  our  figure  will  be  that  of  a  portion 
of  an  unduloid.  If  we  increase  the  pressure  between  the  discs, 
the  peripheral  surface  of  oil  will  bulge  out  more  and  more,  and 
will  presently  constitute  a  portion  of  a  sphere.  But  we  may 
continue  the  process  yet  further,  and  within  certain  limits  we  shall 
find  that  the  system  remains  perfectly  stable.  What  is  this  new 
curved  surface  which  has  arisen  out  of  the  sphere,  as  the  latter 
was  produced  from  the  unduloid  ?  It  is  no  other  than  a  portion 
of  a  nodoid,  that  part  which  in  Fig.  66  lies  between  such  limits  as 
M  and  N.  But  this  surface,  which  is  concave  in  both  directions 
towards  the  surface  of  the  oil  within,  is  exerting  a  pressure  upon 
the  latter,  just  as  did  the  sphere  out  of  which  a  moment  ago  it 
was  transformed ;  and  we  had  just  stated,  in  considering  the 
previous  experiment,  that  the  pressure  inwards  exerted  by  the 
nodoid  was  a  negative  one.  The  explanation  of  this  seeming 
discrepancy  lies  in  the  simple  fact  that,  if  we  follow  the  outline 


V]  OF   FIGURES   OF   EQUILIBRIUM  225 

of  our  nodoid  curve  in  Fig.  66  from  o,  p,  the  surface  concerned 
in  the  former  case,  to  M,  N,  that  concerned  in  the  present,  we  shall 
see  that  in  the  two  experiments  the  surface  of  the  hquid  is  not 
homologous,  but  hes  on  the  positive  side  of  the  curve  in  the  one 
case  and  on  the  negative  side  in  the  other. 

Of  all  the  surfaces  which  we  have  been  describing,  the  sphere 
is  the  only  one  which  can  enclose  space ;  the  others  can  only  help 
to  do  so,  in  combination  with  one  another  or  with  the  sphere  itself. 
Thus  we  have  seen  that,  in  normal  equiUbrium,  the  cylindrical 
vesicle  is  closed  at  either  end  by  a  portion  of  a  sphere,  and  so  on. 
Moreover  the  sphere  is  not  only  the  only  one  of  our  figures  which 
can  enclose  a  finite  space ;  it  is  also,  of  all  possible  figures,  that 
which  encloses  the  greatest  volume  with  the  least  area  of  surface ; 
it  is  strictly  and  absolutely  the  surface  of  minimal  area,  and  it 
is  therefore  the  form  which  will  be  naturally  assumed  by  a  uni- 
cellular organism  (just  as  by  a  raindrop),  when  it  is  practically 
homogeneous  and  when,  like  Orbulina  floating  in  the  ocean,  its 
surroundings  are  likewise  practically  homogeneous  and  sym- 
metrical. It  is  only  relatively  speaking  that  all  the  rest  are 
surfaces  minimae  areae ;  they  are  so,  that  is  to  say,  under  the 
given  conditions,  which  involve  various  forms  of  pressure  or 
restraint.  Such  restraints  are  imposed,  for  instance,  by  the 
pipes  or  annuli  with  the  help  of  which  we  draw  out  our  cylindrical 
or  unduloid  oil-globule  or  soap-bubble ;  and  in  the  case  of  the 
organic  cell,  similar  restraints  are  constantly  suppUed  by  solidifica- 
tion, partial  or  complete,  local  or  general,  of  the  cell-wall. 

Before  we  pass  to  biological  illustrations  of  our  surface-tension 
figures,  we  have  still  another  preliminary  matter  to  deal  with. 
We  have  seen  from  our  description  of  two  of  Plateau's  classical 
experiments,  that  at  some  particular  point  one  type  of  surface 
gives  place  to  another;  and  again,  we  know  that,  when  we  draw 
out  our  soap-bubble  into  and  then  beyond  a  cylinder,  there  comes 
a  certain  definite  point  at  which  our  bubble  breaks  in  two,  and 
leaves  us  with  two  bubbles  of  which  each  is  a  sphere,  or  a  portion 
of  a  sphere.  In  short  there  are  certain  definite  limits  to  the 
dimensions  of  our  figures,  within  which  limits  equilibrium  is 
stable   but  at  which  it  becomes   unstable,   and  above  which  it 

T.  G.  15 


226 


THE  FORMS   OF   CELLS 


[CH. 


breaks  down.  Moreover  in  our  composite  surfaces,  when  the 
cylinder  for  instance  is  capped  by  two  spherical  cups  or  lenticular 
discs,  there  is  a  well-defined  ratio  which  regulates  their  respective 
curvatures,  and  therefore  their  respective  dimensions.  These  two 
matters  we  may  deal  with  together. 

Let  us  imagine  a  liquid  drop  which  by  appropriate  conditions 
has  been  made  to  assume  the  form  of  a  cylinder ;  we  have  already 
seen  that  its  ends  will  be  terminated  by  portions  of  spheres. 
Since  one  and  the  same  liquid  film  covers  the  sides  and  ends  of 
the  drop  (or  since  one  and  the  same  delicate  membrane  encloses 
the  sides  and  ends  of  the  cell),  we  assume  the  surface-tension  (T) 
to  be  everywhere  identical ;  and  it  follows,  since  the  internal 
fluid-pressure  is  also  everywhere  identical,  that  the  expression 
{1/R  +  l/R')  for  the  cylinder  is  equal  to  the  corresponding  expres- 
sion, which  we  may  call  (1/r  +  1/r'),  in  the  case  of  the  terminal 
spheres.  But  in  the  cylinder  1/R'  =  0,  and  in  the  sphere  1/r  =  1/r'. 
Therefore  our  relation  of  equality  becomes  1/R  =  2/r,  or  r  =  2R; 
that  is  to  say,  the  sphere  in  question  has  just  twice  the  radius  of 
the  cylinder  of  which  it  forms  a  cap. 

And  if  Ob,  the  radius  of  the  sphere,  be  equal  to  twice  the  radius 
(Ort)  of  the  cylinder,  it  follows  that  the  angle  aOb  is  an  angle  of 
,  60°,  and  bOc  is  also  an  angle  of  60° ; 

that  is  to  say,  the  arc  be  is  equal  to 
1 77".  In  other  words,  the  spherical 
disc  which  (under  the  given  conditions) 
caps  our  cylinder,  is  not  a  portion 
taken  at  haphazard,  but  is  neither 
more  nor  less  than  that  portion  of  a 
sphere  which  is  subtended  by  a  cone 
of  60°.  Moreover,  it  is  plain  that 
the    height  of   the  spherical  cap,  de, 


Fig.  67. 


^Ob-ab  =  R{2-^3)^  0-27R, 

where  R  is  the  radius  of  our  cylinder, 
or  one-half  the  radius  of  our  spherical 
cap:  in  other  words  the  normal  height  of  the  spherical  cap  over 
the  end  of  the  cylindrical  cell  is  just  a  very  little  more  than  one- 
eighth  of  the  diameter  of  the  cylinder,  or  of  the  radius  of  the 


V] 


OF   FIGURES   OF  EQUILIBRIUM 


227 


sphere.  And  these  are  the  proportions  which  we  recognise,  ander 
normal  circumstances,  in  such  a  case  as  the  cylindrical  cell  of 
Spirogyra  where  its  free  end  is  capped  by  a  portion  of  a  sphere. 

Among  the  many  important  theoretical  discoveries  which  we 
owe  to  Plateau,  one  to  which  we  have  just  referred  is  of  peculiar 
importance :  namely  that,  with  the  exception  of  the  sphere  and 
the  plane,  the  surfaces  with  which  we  have  been  dealing  are  only 
in  complete  equilibrium  within  certain  dimensional  limits,  or  in 
other  words,  have  a  certain  definite  limit  of  stability ;  only  the  plane 
and  the  sphere,  or  any  portions  of  a  sphere,  are  perfectly  stable, 
because  they  are  perfectly  symmetrical,  figures.  For  experimental 
demonstration,  the  case  of  the  cylinder  is  the  simplest.  If  we 
produce  a  liquid  film  having  the  form  of  a  cylinder,  either  by 


Fig.  68. 


drawing  out  a  bubble  or  by  supporting  between  two  rings  a 
globule  of  oil,  the  experiment  proceeds  easily  until  the  length  of 
the  cylinder  becomes  just  about  three  times  as  great  as  its  diameter. 
But  somewhere  about  this  limit  the  cylinder  alters  its  form ;  it 
begins  to  narrow  at  the  waist,  so  passing  into  an  unduloid,  and 
the  deformation  progresses  quickly  until  at  last  our  cylinder 
breaks  in  two,  and  its  two  halves  assume  a  spherical  form.  It  is 
found,  by  theoretical  considerations,  that  the  precise  limit  of 
stability  is  at  the  point  when  the  length  of  the  cylinder  is  exactly 
equal  to  its  circumference,  that  is  to  say,  when  L  =  IttR,  or  when 
the  ratio  of  length  to  diameter  is  represented  by  tt. 

In  the  case  of  the  catenoid.  Plateau's  experimental  procedure 
was  as  follows.  To  support  his  globule  of  oil  (in,  as  usual,  a 
mixture  of  alcohol  and  water  of  its  own  specific  gravity),  he  used 

15—2      . 


228  THE   FORMS   OF   CELLS  [ch. 

a  pair  of  metal  rings,  which  happened  to  have  a  diameter  of 
71  millimetres;  and,  in  a  series  of  experiments,  he  set  these  rings 
apart  at  distances  of  55,  49,  47,  45,  and  43  mm.  successively. 
In  each  case  he  began  by  bringing  his  oil-globule  into  a  cylindrical 
form,  by  sucking  superfluous  oil  out  of  the  drop  until  this  result 
was  attained ;  and  always,  for  the  reason  with  which  we  are  now 
acquainted,  the  cylindrical  sides  were  associated  with  spherical 
ends  to  the  cylinder.  On  continuing  to  withdraw  oil  in  the  hope 
of  converting  these  spherical  ends  into  planes,  he  found,  naturally, 
that  the  sides  of  the  cylinder  drew  in  to  form  a  concave  surface ; 
but  it  was  by  no  means  easy  to  get  the  extremities  actually  plane : 
and  unless  they  were  so,  thus  indicating  that  the  surface-pressure 
of  the  drop  was  nil,  the  curvature  of  the  sides  could  not  be  that 
of  a  catenoid.  For  in  the  first  experiment,  when  the  rings  were 
55  mm.  apart,  as  soon  as  the  convexity  of  the  ends  was  to  a  certain 
extent  diminished,  it  spontaneously  increased  again ;  and  the 
transverse  constriction  of  the  globule  correspondingly  deepened, 
until  at  a  certain  point  equilibrium  set  in  anew.  Indeed,  the  more 
oil  he  removed,  the  more  convex  became  the  ends,  until  at  last 
the  increasing  transverse  constriction  led  to  the  breaking  of  the- 
oil-globule  into  two.  In  the  third  experiment,  when  the  rings 
were  47  mm.  apart,  it  was  easy  to  obtain  end-surfaces  that  were 
actually  plane,  and  they  remained  so  even  though  more  oil  was 
withdrawn,  the  transverse  constriction  deepening  accordingly. 
Only  after  a  considerable  amount  of  oil  had  been  sucked  up  did 
the  plane  terminal  surface  become  gradually  convex,  and  presently 
the  narrow  waist,  narrowing  more  and  more,  broke  across  in  the 
usual  way.  Finally  in  the  fifth  experiment,  where  the  rings  were 
still  nearer  together,  it  was  again  possible  to  bring  the  ends  of  the 
oil-globule  to  a  plane  surface,  as  in  the  third  and  fourth  experiments, 
and  to  keep  this  surface  plane  in  spite  of  some  continued  with- 
drawal of  oil.  But  very  soon  the  ends  became  gradually  concave, 
and  the  concavity  deepened  as  more  and  more  oil  was  withdrawn, 
until  at  a  certain  limit,  th'e  whole  oil-globule  broke  up  in  general 
disruption. 

We  learn  from  this  that  the  limiting  size  of  the  catenoid  was 
reached  when  the  distance  of  the  supporting  rings  was  to  their 
diameter  as  47  to  71,  or,  as  nearly  as  possible,  as  two  to  three; 


V]  OF  FIGURES   OF   EQUILIBRIUM  229 

and  as  a  matter  of  fact  it  can  be  shewn  that  2/3  is  the  true 
theoretical  value.  Above  this  limit  of  2/3,  the  inevitable  convexity 
of  the  end-surfaces  shows  that  a  positive  pressure  inwards  is  being 
exerted  by  the  surface  film,  and  this  teaches  us  that  the  sides  of 
the  figure  actually  constitute  not  a  catenoid  but  an  unduloid, 
whose  spontaneous  changes  tend  to  a  form  of  greater  stability. 
Below  the  2/3  limit  the  catenoid  surface  is  essentially  unstable, 
and  the  form  into  which  it  passes  under  certain  conditions  of 
disturbance  such  as  that  of  the  excessive  withdrawal  of  oil,  is 
that  of  a  nodoid  (Fig.  65  a). 

The  unduloid  has  certain  peculiar  properties  as  regards  its 
limitations  of  stability.  But  as  to  these  we  need  mention  two 
facts  only:  (1)  that  when  the  unduloid,  which  we  produce  with 
our  soap-bubble  or  our  oil-globule,  consists  of  the  figure  containing 
a  complete  constriction,  it  has  somewhat  wide  limits  of  stability ; 
but  (2)  if  it  contain  the  swollen  portion,  then  equilibrium  is  limited 
to  the  condition  that  the  figure  consists  simply  of  one  complete 
unduloid,  that  is  to  say  that  its  ends  are  constituted  by  the 
narrowest  portions,  and  its  middle  by  the  widest  portion  of  the 
entire  curve.  The  theoretical  proof  of  this  latter  fact  is  difiicult, 
but  if  we  take  the  proof  for  granted,  the  fact  will  serve  to  throw 
light  on  what  we  have  learned  regarding  the  stability  of  the  cylinder. 
For,  when  we  remember  that  the  meridional  section  of  our  unduloid 
is  generated  by  the  rolling  of  an  ellipse  upon  a  straight  line  in  its 
own  plane,  we  shall  easily  see  that  the  length  of  the  entire  unduloid 
is  equal  to  the  circumference  of  the  generating  ellipse.  As  the 
unduloid  becomes  less  and  less  sinuous  in  outline,  it  gradually 
approaches,  and  in  time  reaches,  the  form  of  a  cylinder;  and 
correspondingly,  the  ellipse  which  generated  it  has  its  foci  more 
and  more  approximated  until  it  passes  into  a  circle.  The  cylinder 
of  a  length  equal  to  the  circumference  of  its  generating  circle  is 
therefore  precisely  homologous  to  an  unduloid  whose  length  is 
equal  to  the  circumference  of  its  generating  ellipse;  and  this  is 
just  what  we  recognise  as  constituting  one  complete  segment  of 
the  unduloid. 

While  the  figures  of  equihbrium  which  are  at  the  same  time 
surfaces  of  revolution  are  only  six  in  number,  there  is  an  infinite 


230  THE   FOEMS   OF  CELLS  [ch. 

number  of  figures  of  equilibrium,  that  is  to  say  of  surfaces  of 
constant  mean  curvature,  which  are  not  surfaces  of  revolution ; 
and  it  can  be  shewn  mathematically  that  any  given  contour  can 
be  occupied  by  a  finite  portion  of  some  one  such  surface,  in  stable 
equilibrium.  The  experimental  verification  of  this  theorem  lies  in 
the  simple  fact  (already  noted)  that  however  we  may  bend  a  wire 
into  a  closed  curve,  plane  or  not  plane,  we  may  always,  under 
appropriate  precautions,  fill  the  entire  area  with  an  unbroken 
film. 

Of  the  regular  figures  of  equilibrium,  that  is  to  say  surfaces 
of  constant  mean  curvature,  apart  from  the  surfaces  of  revolution 
which  we  have  discussed,  the  helicoid  spiral  is  the  most  interesting 
to  the  biologist.  This  is  a  helicoid  generated  by  a  straight  line 
perpendicular  to  an  axis,  about  which  it  turns  at  a  uniform  rate 
while  at  the  same  time  it  slides,  also  uniformly,  along  this  same 
axis.  At  any  point  in  this  surface,  the  curvatures  are  equal  and 
of  opposite  sign,  and  the  sum  of  the  curvatures  is  accordingly  nil. 
Among  what  are  called  "ruled  surfaces"  (which  we  may  describe 
as  surfaces  capable  of  being  defined  by  a  system  of  stretched 
strings),  the  plane  and  the  helicoid  are  the  only  two  whose  mean 
curvature  is  null,  while  the  cylinder  is  the  only  one  whose  curvature 
is  finite  and  constant.  As  this  simplest  of  helicoids  corresponds, 
in  three  dimensions,  to  what  in  two  dimensions  is  merely  a  plane 
(the  latter  being  generated  by  the  rotation  of  a  straight  line  about 
an  axis  without  the  superadded  gliding  motion  which  generates 
the  helicoid),  so  there  are  other  and  much  more  complicated 
helicoids  which  correspond  to  the  sphere,  the  unduloid  and  the 
rest  of  our  figures  of  revolution,  the  generating  planes  of  these 
latter  being  supposed  to  wind  spirally  about  an  axis.  In  the  case 
of  the  cylinder  it  is  obvious  that  the  resulting  figure  is  indistinguish- 
able from  the  cylinder  itself.  In  the  case  of  the  unduloid  we 
obtain  a  grooved  spiral,  such  as  we  may  meet  with  in  nature  (for 
instance  in  Spirochsetes,  Bodo  gracilis,  etc.),  and  which  accordingly 
it  is  of  interest  to  us  to  be  able  to  recognise  as  a  surface  of  minimal 
area  or  constant  curvature. 

The  foregoing  considerations  deal  with  a  small  part  only 
of  the  theory  of  surface  tension,  or  of  capillarity:  with  that 
part,  namely,  which  relates  to  the  forms  of  surface  which  are 


V]  OF  FIGURES   OF   EQUILIBRIUM  231 

capable  of  subsisting  in  equilibrium  under  the  action  of  that  force, 
either  of  itself  or  subject  to  certain  simple  constraints.  And  as 
yet  we  have  limited  ourselves  to  the  case  of  a  single  surface,  or 
of  a  single  drop  or  bubble,  leaving  to  another  occasion  a  discussion 
of  the  forms  assumed  when  such  drops  or  vesicles  meet  and  com- 
bine together.  In  short,  what  we  have  said  may  help  us  to  under- 
stand the  form  of  a  cell, — considered,  as  with  certain  limitations 
we  may  legitimately  consider  it,  as  a  liquid  drop  or  liquid  vesicle ; 
the  conformation  of  a  tissue  or  cell-aggregate  must  be  dealt  with 
in  the  light  of  another  series  of  theoretical  considerations.  In 
both  cases,  we  can  do  no  more  than  touch  upon  the  fringe  of  a 
large  and  difficult  subject.  There  are  many  forms  capable  of 
realisation  under  surface  tension,  and  many  of  them  doubtless  to 
be  recognised  among  organisms,  which  we  cannot  touch  upon  in 
this  elementary  account.  The  subject  is  a  very  general  one;  it 
is,  in  its  essence,  more  mathematical  than  physical ;  it  is  part  of 
the  mathematics  of  surfaces,  and  only  comes  into  relation  with 
surface  tension,  because  this  physical  phenomenon  illustrates  and 
exemplifies,  in  a  concrete  way,  most  of  the  simple  and  symmetrical 
conditions  with  which  the  general  mathematical  theory  is  capable 
of  dealing.  And  before  we  pass  to  illustrate  by  biological  examples 
the  physical  phenomena  which  we  have  described,  we  must  be 
careful  to  remember  that  the  physical  conditions  which  we  have 
hitherto  presupposed  will  never  be  wholly  realised  in  the  organic 
cell.  Its  substance  will  never  be  a  perfect  fluid,  and  hence 
equilibrium  will  be  more  or  less  slowly  reached;  its  surface  will 
seldom  be  perfectly  homogeneous,  and  therefore  equilibrium  will 
(in  the  fluid  condition)  seldom  be  perfectly  attained ;  it  will  very 
often,  or  generally,  be  the  seat  of  other  forces,  symmetrical  or 
unsymmetrical ;  and  all  these  causes  will  more  or  less  perturb  the 
effects  of  surface  tension  acting  by  itself.  But  we  shall  find  that, 
on  the  whole,  these  effects  of  surface  tension  though  modified  are 
not  obliterated  nor  even  masked ;  and  accordingly  the  phenomena 
to  which  I  have  devoted  the  foregoing  pages  will  be  found 
manifestly  recurring  and  repeating  themselves  among  the  pheno- 
mena of  the  organic  cell. 

In  a  spider's  web  we  find  exemplified  several  of  the  principles 


232  THE   FORMS   OF   CELLS  [ch. 

of  surface  tension  which  we  have  now  explained.  The  thread  is 
formed  out  of  the  fluid  secretion  of  a  gland,  and  issues  from  the 
body  as  a  semi-fluid  cylinder,  that  is  to  say  in  the  form  of  a  surface 
of  equilibrium,  the  force  of  expulsion  giving  it  its  elongation  and 
that  of  surface  tension  giving  it  its  circular  section.  It  is  prevented, 
by  almost  immediate  solidification  on  exposure  to  the  air,  from 
breaking  up  into  separate  drops  or  spherules,  as  it  would  otherwise 
tend  to  do  as  soon  as  the  length  of  the  cyhnder  had  passed  its 
limit  of  stabihty.  But  it  is  otherwise  with  the  sticky  secretion 
which,  coming  from  another  gland,  is  simultaneously  poured  over 
the  issuing  thread  when  it  is  to  form  the  spiral  portion  of  the 
web.  This  latter  secretion  is  more  fluid  than  the  first,  and  retains 
its  fluidity  for  a  very  much  longer  time,  finally  drying  up  after 
several  hours.  By  capillarity  it  "wets"  the  thread,  spreading 
itself  over  it  in  an  even  film,  which  film  is  now  itself  a  cylinder. 
But  this  liquid  cylinder  has  its  limit  of  stability  when  its  length 
equals  its  own  circumference,  and  therefore  just  at  the  points  so 
defined  it  tends  to  disrupt  into  separate  segments :  or  rather,  in 
the  actual  case,  at  points  somewhat  more  distant,  owing  to  the 
imperfect  fluidity  of  the  viscous  film,  and  still  more  to  the  frictional 
drag  upon  it  of  the  inner  solid  cylinder,  or  thread,  with  which  it 
is  in  contact.  The  cylinder  disrupts  in  the  usual  manner,  passing 
first  into  the  wavy  outline  of  an  unduloid,  whose  swollen  portions 
swell  more  and  more  till  the  contracted  parts  break  asunder,  and 
we  arrive  at  a  series  of  spherical  drops  or  beads,  of  equal  size, 
strung  at  equal  intervals  along  the  thread.  If  we  try  to  spread 
varnish  over  a  thin  stretched  wire,  we  produce  automatically  the 
same  identical  result  *  ;  unless  our  varnish  be  such  as  to  dry  almost 
instantaneously,  it  gathers  into  beads,  and  do  what  we  can,  we 
fail  to  spread  it  smooth.  It  follows  that,  according  to  the  viscidity 
and  drying  power  of  the  varnish,  the  process  may  stop  or  seem  to 
stop  at  any  point  short  of  the  formation  of  the  perfect  spherules ; 
it  is  quite  possible,  therefore,  that  as  our  final  stage  we  may  only 
obtain  half-formed  beads,  or  the  wavy  outline  of  an  unduloid. 
The  formation  of  the  beads  may  be  facilitated  or  hastened  by 
jerking  the  stretched  thread,  as  the  spider  actually  does:    the 

*  Felix  Plateau  recommends  the  use  of  a  weighted  thread,  or  plumb-line, 
drawn  up  out  of  a  jar  of  water  or  oil;    Phil.  Mag.  xxxiv,  p.  246,  1867. 


V]  OF   SPIDERS'   WEBS  233 

effect  of  the  jerk  being  to  disturb  and  destroy  the  unstable 
equiHbriiim  of  the  viscid  cyhnder*.  Another  very  curious 
phenomenon   here   presents  itself. 

In  Plateau's  experimental  separation  of  a  cyhnder  of  oil  into 
two  spherical  portions,  it  was  noticed  that,  when  contact  was 
nearly  broken,  that  is  to  say  when  the  narrow  neck  of  the  unduloid 
had  become  very  thin,  the  two  spherical  bullae,  instead  of  absorbing 
the  fluid  out  of  the  narrow  neck  into  themselves  as  they  had  done 
with  the  preceding  portion,  drew  out  this  small  remaining  part  of 
the  liquid  into  a  thin  thread  as  they  completed  their  spherical 
form  and  consequently  receded  from  one  another :  the  reason  being 
that,  after  the  thread  or  "neck"  has  reached  a  certain  tenuity, 
the  internal  friction  of  the  fluid  prevents  or  retards  its  rapid  exit 
from  the  little  thread  to  the  adjacent  spherule.  It  is  for  the  samfe 
reason  that  we  are  able  to  draw  a  glass  rod  or  tube,  which  we  have 
heated  in  the  middle,  into  a  long  and  uniform  cylinder  or  thread, 
by  quickly  separating  the  two  ends.  But  in  the  case  of  the  glass 
rod,  the  long  thin  intermediate  cylinder  quickly  cools  and  solidifies, 
while  in  the  ordinary  separation  of  a  liquid  cylinder  the  corre- 
sponding intermediate  cylinder  remains  liquid ;  and  therefore,  like 
any  other  liquid  cylinder,  it  is  liable  to  break  up,  provided  that  its 
dimensions  exceed  the  normal  limit  of  stability.  And  its  length 
is  generally  such  that  it  breaks  at  two  points,  thus  leaving  two 
terminal  portions  continuous  with  the  spheres  and  becoming 
confluent  with  these,  and  one  median  portion  w^hich  resolves  itself 
into  a  comparatively  tiny  spherical  drop,  midway  between  the 
original  and  larger  two.  Occasionally,  the  same  process  of  forma- 
tion of  a  connecting  thread  repeats  itself  a  second  time,  between 
the  small  intermediate  spherule  and  the  large  spheres ;  and  in  this 
case  we  obviously  obtain  two  additional  spherules,  still  smaller  in 
size,  and  lying  one  on  either  side  of  our  first  little  one.  This  whole 
phenomenon,  of  equal  and  regularly  interspaced  beads,  often  with 
little  beads  regularly  interspaced  between  the  larger  ones,  and 
possibly  also  even  a  third  series  of  still  smaller  beads  regularly 
intercalated,  may  be  easily  observed  in  a  spider's  web,  such  as 
that   of    Epeira,    very   often    with    beautiful    regularity, — which 

*  Cf.  Boys,  C.  v.,  On  Quartz  Fibres,  Nature,  July  11,  1889;  Warburton,  C, 
The  Spinning  Apparatus  of  Geometric  Spiders,  Q.J. M.S.  xxxi,  pp.  29-39,  1890. 


234  THE   FORMS   OF   CELLS  [ch. 

naturally,  however,  is  sometimes  interrupted  and  disturbed  owing 
to  a  slight  want  of  homogeneity  in  the  secreted  fluid ;  and  the 
same  phenomenon  is  repeated  on  a  grosser  scale  when  the  web  is 
bespangled  with  dew,  and  every  thread  bestrung  with  pearls 
innumerable.  To  the  older  naturalists,  these  regularly  arranged 
and  beautifully  formed  globules  on  the  spider's  web  were  a  cause 
of  great  wonder  and  admiration.  Blackwall,  counting  some 
twenty  globules  in  a  tenth  of  an  inch,  calculated  that  a  large 
garden-spider's  web  comprised  about  120,000  globules ;  the  net 
was  spun  and  finished  in  about  forty  minutes,  and  Blackwall  was 
evidently  filled  with  astonishment  at  the  skill  and  quickness  with 
which  the  spider  manufactured  these  little  beads.  And  no  wonder, 
for  according  to  the  above  estimate  they  had  to  be  made  at  the 
rate  of  about  50  per  second*. 

The  little  delicate  beads  which  stud  the  long  thin  pseudopodia 
of  a  foraminifer,  such  as  Gromia,  or  which  in  like  manner  appear 


Fig.  69.     Hair  of  Tnanea,  in  gtycerine.     (After  Berthold.) 


upon  the  cylindrical  film  of  protoplasm  which  covers  the  long 
radiating  spicules  of  6'^o6i^e>ma,  represent  an  identical  phenomenon. 
Indeed  there  are  many  cases,  in  which  we  may  study  in  a  proto- 
plasmic filament  the  whole  process  of  formation  of  such  beads. 
If  we  squeeze  out  on  to  a  slide  the  viscid  contents  of  a  mistletoe 
berry,  the  long  sticky  threads  into  which  the  substance  runs  shew 
the  whole  phenomenon  particularly  well.  Another  way  to 
demonstrate  it  was  noticed  many  years  ago  by  Hofmeister  and 
afterwards  explained  by  Berthold.  The  hairs  of  certain  water- 
plants,  such  as  Hydrocharis  or  Trianea,  constitute  very  long  cylin- 
drical cells,  the  protoplasm  being  supported,  and  maintained  in 
equilibrium  by  its  contact  with  the  cell-wall.  But  if  we  immerse 
the  filament  in  some  dense  fluid,  a  little  sugar-solution  for  instance, 
or  dilute  glycerine,  the  cell-sap  tends  to  diffuse  outwards,  the  proto- 
plasm parts  company  with  its  surrounding  and  supporting  wall, 

*  J.  Blackwall,  Spiders  of  Great  Britain  (Ray  Society),  1859,  p.   10;    Trans. 
Linn.  Soc.  xvi,  p.  477,  1833. 


V] 


OF   GLOBULES   OR   BEADS 


235 


and  lies  free  as  a  protoplasmic  cylinder  in  the  interior  of  the  cell. 
Thereupon  it  immediately  shews  signs  of  instability,  and  commences 
to  disrupt.  It  tends  to  gather  into  -spheres,  which  however,  as  in 
our  illustration,  may  be  prevented  by  their  narrow  quarters  from 
assuming  the  complete  spherical  form ;  and  in  between  these 
spheres,  we  have  more  or  less  regularly  alternate  ones,  of  smaller 
size*.  Similar,  but  less  regular,  beads  or  droplets  may  be  caused  to 
appear,  under  stimulation  by  an  alternating  current,  in  the  proto- 
plasmic threads  within  the  living  cells  of  the  hairs  of  Tradescantia. 
The  explanation  usually  given  is,  that  the  viscosity  of  the  proto- 


Fig.  70.     Phases  of  a  Splash.     (From  Worthington. ) 

plasm  is  reduced,  or  its  fluidity  increased ;    but  an  increase  of  the 
surface  tension  would  seem  a  more  likely  reason  f. 


We  may  take  note  here  of  a  remarkable  series  of  phenomena, 
which,  though  they  seem  at  first  sight  to  be  of  a  very  different 
order,  are  closely  related  to  the  phenomena  which  attend  and 
which  bring  about  the  breaking-up  of  a  liquid  cylinder  or  thread. 

In  some  of  Mr  Worthington's  most  beautiful  "experiments  on 

*  The  intermediate  spherules  appear,  with  great  regularity  and  beauty,  whenever 
a  hquid  jet  breaks  up  into  drops;  see  the  instantaneous  photographs  in  Poynting 
and  Thomson's  Proiierties  of  Matter,  pp.   151,  152,  (ed.   1907). 

■]■  Kiihne,  Untersiwhungen  iiber  das  Protoplasma,  1864,  p.  75,  etc. 


236 


THE   FORMS   OF  CELLS 


[CH. 


splashes,  it  was  found  that  the  fall  of  a  round  pebble  into  water 
from  a  considerable  height,  caused  the  rise  of  a  filmy  sheet  of  water 
in  the  form  of  a  cup  or  cylinder ;  and  the  edge  of  this  cylindrical 
film  tended  to  be  cut  up  into  alternate  lobes  and  notches,  and  the 
prominent  lobes  or  "jets"  tended,  in  more  extreme  cases,  to  break 
off  or  to  break  up  into  spherical  beads  (Fig.  70)*.  A  precisely 
similar  appearance  is  seen,  on  a  great  scale,  in  the  thin  edge  of  a 
breaking  wave :  when  the  smooth  cylindrical  edge,  at  a  given 
moment,  shoots  out  an  array  of  tiny  jets  which  break  up  into 
the  droplets  which  constitute  "  spray "  (Fig.  71,  a,  b).  We 
are  at  once  reminded  of  the  beautifully  symmetrical  notching  on 
the  calycles  of  many  hydroids,  which  little  cups  before  they  became 
stiff  and  rigid  had  begun  their  existence  as  liquid  or  semi-liquid 
films. 


Fig.  71.     A  breaking  wave.     (From  Worthington. ) 

The  phenomenon  is  two-fold.  In  the  first  place,  the  edge  of 
our  tubular  or  crater-like  film  forms  a  liquid  ring  or  annulus, 
which  is  closely  comparable  with  the  liquid  thread  or  cylinder 
which  we  have  just  been  considering,  if  only  we  conceive  the  thread 
to  be  bent  round  into  the  ring.  And  accordingly,  just  as  the  thread 
spontaneously  segments,  first  into  an  unduloid,  and  then  into 
separate  spherical  drops,  so  likewise  will  the  edge  of  our  annulus 
tend  to  do.  This  phase  of  notching,  or  beading,  of  the  edge  of 
the  film  is  beautifully  seen  in  many  of  Worthington's  experiments'}". 
In  the  second  place,  the  very  fact  of  the  rising  of  the  crater  means 
that  liquid  is  flowing  up  from  below  towards  the  rim ;  and  the 
segmentation  of  the  rim  means  that  channels  of  easier  flow  are 

*  A  Study  of  Splashes,  1908,  p.  38,  etc. ;  Segmentation  of  a  Liquid  Annulus, 
Proc.  Roy.  Soc.  xxx,  pp.  49-60,  1880. 

f  Cf.  ibid.  pp.  17,  77.  The  same  phenomenon  is  beautifully  and  continuously 
•evident  when  a  strong  jet  of  water  from  a  tap  impinges  on  a  curved  surface  and  then 
shoots  off  it. 


V] 


THE   SHAPE   OF   A  SPLASH 


•237 


created,  along  which  the  Hquid  is  led,  or  is  driven,  into  the  pro- 
tuberances: and  these  are  thus  exaggerated  into  the  jets  or  arms 
which  are  sometimes  so  conspicuous  at  the  edge  of  the  crater. 
In  short,  any  film  or  film-like  cup,  fluid  or  semi-fluid  in  its  consis- 
tency, will,  like  the  straight  liquid  cylinder,  be  unstable :  and  its 
instability  will  manifest  itself  (among  other  ways)  in  a  tendency 
to  segmentation  or  notching  of  the  edge ;  and  just  such  a  peripheral 
notching  is  a  conspicuous  feature  of  many  minute  organic  cup-like 
structures.  In  the  case  of  the  hydroid  calycle  (Fig.  72),  we  are  led 
to  the  conclusion  that  the  two  common  and  conspicuous  features 
of  notching  or  indentation  of  the  cup,  and  of  constriction  or 
annulation  of  the  long  cylindrical  stem,  are  phenomena  of  the 
same  order  and  are  due  to  surface-tension  in  both  cases  alike. 


Fig.  72.     Calycles  of  Campanularian  zoophytes.     (A)   C.  Integra; 
(B)   C.  groenlandica ;    (C)    C.  bispinosa;    (D)    C.  raridentafa. 


Another  phenomenon  displayed  in  the  same  experiments  is  the 
formation  of  a  rope-like  or  cord-like  thickening  of  the  edge  of  the 
annulus.  This  is  due  to  the  more  or  less  sudden  checking  at  the 
rim  of  the  flow  of  liquid  rising  from  below :  and  a  similar  peri|)heral 
thickening  is  frequently  seen,  not  only  in  some  of  our  hydroid 
cups,  but  in  many  Vorticellas  (cf.  Fig.  75),  and  other  organic 
cup-like  conformations.  A  perusal  of  Mr  Worthington's  book 
will  soon  suggest  that  these  are  not  the  only  manifestations  of 
surface-tension  in  connection  with  splashes  which  present  curious 
resemblances  and  analogies  to  phenomena  of  organic  form. 

The  phenomena  of  an  ordinary  liquid  splash  are  so  swiftly 


238  THE  FORMS   OF  CELLS  [cii. 

transitory  that  their  study  is  only  rendered  possible  by  "instan- 
taneous" photography:  but  this  excessive  rapidity  is  not  an 
essential  part  of  the  phenomenon.  For  instance,  we  can  repeat 
and  demonstrate  many  of  the  simpler  phenomena,  in  a  permanent 
or  quasi-permanent  form,  by  splashing  water  on  to  a  surface  of 
dry  sand,  or  by  firing  a  bullet  into  a  soft  metal  target.  There  is 
nothing,  then,  to  prevent  a  slow  and  lasting  manifestation,  in 
a  viscous  medium  such  as  a  protoplasmic  organism,  of  phenomena 
which  appear  and  disappear  with  prodigious  rapidity  in  a  more 
mobile  liquid.  Nor  is  there  anything  peculiar  in  the  "splash" 
itself;  it  is  simply  a  convenient  method  of  setting  up  certain 
niotions  or  currents,  and  producing  certain  surface-forms,  in  a 
liquid  medium, — or  even  in  such  an  extremely  imperfect  fluid  as 
is  represented  (in  another  series  of  experiments)  by  a  bed  of  sand. 
Accordingly,  we  have  a  large  range  of  possible  conditions  under 
which  the  organism  might  conceivably  display  configurations 
analogous  to,  or  identical  with,  those  which  Mr  Worthington  has 
shewn  us  how  to  exhibit  by  one  particular  experimental  method. 
To  one  who  has  watched  the  potter  at  his  wheel,  it  is  plain 
that  the  potter's  thumb,  like  the  glass-blower's  blast  of  air, 
depends  for  its  efficacy  upon  the  physical  properties  of  the 
medium  on  which  it  operates,  which  for  the  time  being  is  essentially 
a  fluid.  The  cup  and  the  saucer,  like  the  tube  and  the  bulb, 
display  (in  their  simple  and  primitive  forms)  beautiful  surfaces  of 
equilibrium  as  manifested  under  certain  limiting  conditions. 
They  are  neither  more  nor  less  than  glorified  "splashes,"  formed 
slowly,  under  conditions  of  restraint  which  enhance  or  reveal 
their  mathematical  symmetry.  We  have  seen,  and  we  shall  see 
again  before  we  are  done,  that  the  art  of  the  glass-blower  is  full 
of  lessons  for  the  naturalist  as  also  for  the  physicist:  illustrating 
as  it  does  the  development  of  a  host  of  mathematical  configura- 
tions and  organic  conformations  which  depend  essentially  on  the 
establishment  of  a  constant  and  uniform  pressure  within  a  closed 
elastic  shell  or  fluid  envelope.  In  like  manner  the  potter's  art 
illustrates  the  somewhat  obscurer  and  more  complex  problems 
(scarcely  less  frequent  in  biology)  of  a  figure  of  equilibrium  which 
is  an  ofen  surface,  or  solid,  of  revolution.  It  is  clear,  at  the  same 
time,  that  the  two  series  of  problems  are  closely  akin;    for  the 


V]  THE   SHAPE   OF  A  SPLASH  239 

glass-blower  can  make  most  things  that  the  potter  makes,  by 
cutting  oft'  'portions  of  his  hollow  ware.  And  besides,  when  this 
fails,  and  the  glass-blower,  ceasing  to  blow,  begins  to  use  his  rod 
to  trim  the  sides  or  turn  the  edges  of  wineglass  or  of  beaker,  he 
is  merely  borrowing  a  trick  from  the  craft  of  the  potter. 

It  would  be  venturesome  indeed  to  extend  our  comparison 
with  these  liquid  surface-tension  phenomena  from  the  cup  or 
calycle  of  the  hydrozoon  to  the  little  hydroid  polype  within :  and 
yet  I  feel  convinced  that  there  is  something  to  be  learned  by  such 
a  comparison,  though  not  without  much  detailed  consideration 
and  mathematical  study  of  the  surfaces  concerned.  The  cylin- 
drical body  of  the  tiny  polype,  the  jet-like  row  of  tentacles,  the 
beaded  annulations  which  these  tentacles  exhibit,  the  web-like 
film  which  sometimes  (when  they  stand  a  little  way  apart)  conjoins 
their  bases,  the  thin  annular  film  of  tissue  which  surrounds  the 
little  organism's  mouth,  and  the  manner  in  which  this  annular 
"peristome"  contracts*,  like  a  shrinking  soap-bubble,  to  close  the 
aperture,  are  every  one  of  them  features  to  which  we  may  find 
a  singular  and  striking  parallel  in  the  surface-tension  phenomena 
which  Mr  Worthington  has  illustrated  and  demonstrated  in  the 
case  of  the  splash. 

Here  however,  we  may  freely  confess  that  we  are  for  the 
present  on  the  uncertain  ground  of  suggestion  and  conjecture; 
and  so  must  we  remain,  in  regard  to  many  other  simple  and 
symmetrical  organic  forms,  until  their  form  and  dynamical 
stability  shall  have  been  investigated  by  the  mathematician :  in 
other  words,  until  the  mathematicians  shall  have  become  persuaded 
that  there  is  an  immense  unworked  field  wherein  they  may  labour, 
in  the  detailed  study  of  organic  form. 

According  to  Plateau,  the  viscidity  of  the  liquid,  while  it 
helps  to  retard  the  breaking  up  of  the  cyhnder  and  so  increases 
the  length  of  the  segments  beyond  that  which  theory  demands, 
has  nevertheless  less  influence  in  this  direction  than  we  might 
have  expected.  On  the  other  hand,  any  external  support  or 
adhesion,  such  as  contact  with  a  solid  body,  will  be  equivalent  to 
a  reduction  of  surface-tension  and  so  will  very  greatly  increase  the 
*  See  a  Study  of  Splashes,  p.  54. 


240  THE   FORMS   OF  CELLS  [ch. 

stability  of  our  cylinder.  It  is  for  this  reason  that  the  mercury 
in  our  thermometer  tubes  does  not  as  a  rule  separate  into  drops, 
though  it  occasionally  does  so,  much  to  our  inconvenience.  And 
again  it  is  for  this  reason  that  the  protoplasm  in  a  long  and  growing 
tubular  or  cylindrical  cell  does  not  necessarily  divide  into  separate 
cells  and  internodes,  until  the  length  of  these  far  exceeds  the 
theoretic  limits.  Of  course  however  and  whenever  it  does  so,  we 
must,  without  ever  excluding  the  agency  of  surface  tension, 
remember  that  there  may  be  other  forces  affecting  the  latter,  and 
accelerating  or  retarding  that  manifestation  of  surface  tension  by 
which  the  cell  is  actually  rounded  off  and  divided. 

In  most  liquids.  Plateau  asserts  that,  on  the  average,  the 
influence  of  viscosity  is  such  as  to  cause  the  cylinder  to  segment 
when  its  length  is  about  four  times,  or  at  most  from  four  to  six 
times  that  of  its  diameter :  instead  of  a  fraction  over  three  times 
as,  in  a  perfect  fluid,  theory  would  demand.  If  we  take  it  at 
four  times,  it  may  then  be  shewn  that  the  resulting  spheres  would 
have  a  diameter  of  about  1-8  times,  and  their  distance  apart  would 
be  equal  to  about  2-2  times  the  diameter  of  the  original  cylinder. 
The  calculation  is  not  difficult  which  would  shew  how  these 
numbers  are  altered  in  the  case  of  a  cylinder  formed  around  a  solid 
core,  as  in  the  case  of  the  spider's  web.  Plateau  has  also  made 
the  interesting  observation  that  the  time  taken  in  the  process  of 
division  of  the  cylinder  is  directly  proportional  to  the  diameter 
of  the  cylinder,  while  varying  considerably  with  the  nature  of  the 
liquid.  This  question,  of  the  time  occupied  in  the  division  of  a 
cell  or  filament,  in  relation  to  the  dimensions  of  the  latter,  has  not 
so  far  as  I  know  been  enquired  into  by  biologists. 

From  the  simple  fact  that  the  sphere  is  of  all  surfaces  that 
whose  surface-area  for  a  given  volume  is  an  absolute  minimum, 
we  have  already  seen  it  to  be  plain  that  it  is  the  one  and  only 
figure  of  equilibrium  which  will  be  assumed  under  surface-tension 
by  a  drop  or  vesicle,  when  no  other  disturbing  factors  are  present. 
One  of  the  most  important  of  these  disturbing  factors  will  be 
introduced,  in  tjie  form  of  complicated  tensions  and  pressures, 
when  one  drop  is  in  contact  with  another  drop  and  when  a  system 
of  intermediate  films  or  partition  walls  is  developed  between  them. 


V]  OF  ASYMMETRY  AND   ANISOTROPY  241 

This  subject  we  shall  discuss  later,  in  connection  with  cell- 
aggregates  or  tissues,  and  we  shall  find  that  further  theoretical 
considerations  are  needed  as  a  preliminary  to  any  such  enquiry. 
Meanwhile  let  us  consider  a  few  cases  of  the  forms  of  cells,  either 
solitary,  or  in  such  simple  aggregates  that  their  individual  form  is 
little  disturbed  thereby. 

Let  us  clearly  understand  that  the  cases  we  are  about  to 
consider  are  those  cases  where  the  perfect  symmetry  of  the  sphere 
is  replaced  by  another  symmetry,  less  complete,  such  as  that  of 
an  elhpsoidal  or  cyUndrical  cell.  The  cases  of  asymmetrical 
deformation  or  displacement,  such  as  is  illustrated  in  the  production 
of  a  bud  or  the  development  of  a  lateral  branch,  are  much  simpler. 
For  here  we  need  only  assume  a  sUght  and  locaHsed  variation  of 
surface-tension,  such  as  may  be  brought  about  in  various  ways 
through  the  heterogeneous  chemistry  of  the  cell;  to  this  point 
we  shall  return  in  our  chapter  on  Adsorption.  But  the  diffused 
and  graded  asymmetry  of  the  system,  which  brings  about  for 
instance  the  elhpsoidal  shape  of  a  yeast-cell,  is  another  matter. 

If  the  sphere  be  the  one  surface  of  complete  symmetry  and 

therefore  of  independent  equilibrium,  it  follows  that  in  every  cell 

which  is  otherwise  conformed  there  must  be  some  definite  force 

to  cause  its  departure  from  sphericity ;    and  if  this  cause  be  the 

very  simple  and  obvious  one  of  the  resistance  offered  by  a  solidified 

envelope,  such  as  an  egg-shell  or  firm  cell-wall,  we  must  still  seek 

for  the  deforming  force  which  was  in  action  to  bring  about  the 

given  shape,  prior  to  the  assumption  of  rigidity.     Such  a  cause 

may  be  either  external  to,  or  may  lie  within,  the  cell  itself.     On 

the  one  hand  it  may  be  due  to  external  pressure  or  to  some  form 

of  mechanical  restraint:    as  it  is  in  all  our  experiments  in  which 

we  submit  our  bubble  to  the  partial  restraint  of  discs  or  rings  or 

more  complicated  cages  of  wire ;  and  on  the  other  hand  it  may  be 

due  to  intrinsic  causes,  which  must  come  under  the  head  either  of 

differences   of  internal   pressure,   or   of  lack   of  homogeneity   or 

isotropy  in  the  surface  itself*. 

*  A  case  which  we  have  not  specially  considered,  but  which  may  be  found  to 
deserve  consideration  in  biology,  is  that  of  a  cell  or  drop  suspended  in  a  liquid  of 
varying  density,  for  instance  in  the  upper  layers  of  a  fluid  (e.g.  sea-water)  at  whose 
surface  condensation  is  going  on,  so  as  to  produce  a  steady  density-gradient.  In 
this  case  the  normally  spherical  drop  will  be  flattened  into  an  oval  form,  with  its 

T.  G,  16 


242  THE 'FORMS   OF  CELLS  [ch. 

Our  full  formula  of  equilibrium,  or  equation  to  an  elastic 
surface,  is  P  =  'p^+  {TIE  +  T'/R'),  where  P  is  the  internal 
pressure,  fg  any  extraneous  pressure  normal  to  the  surface,  R,  R' 
the  radii  of  curvature  at  a  point,  and  T,  T' ,  the  corresponding 
tensions,  normal  to  one  another,  of  the  envelope. 

Now  in  any  given  form  which  we  are  seeking  to  account  for, 
R,  R'  are  known  quantities ;  but  all  the  other  factors  of  the  equation 
are  unknown  and  subject  to  enquiry.  And  somehow  or  other,  by 
this  formula,  we  must  account  for  the  form  of  any  solitary  cell 
whatsoever  (provided  always  that  it  be  not  formed  by  successive 
stages  of  solidification),  the  cylindrical  cell  of  Spirogyra,  the 
ellipsoidal  yeast-cell,  or  (as  we  shall  see  in  another  chapter)  the 
shape  of  the  egg  of  any  bird.  In  using  this  formula  hitherto,  we 
have  taken  it  in  a  simplified  form,  that  is  to  say  Ave  have  made 
several  limiting  assumptions.  We  have  assumed  that  P  was 
simply  the  uniform  hydrostatic  pressure,  equal  in  all  directions, 
of  a  body  of  liquid ;  we  have  assumed  that  the  tension  T  was 
simply  due  to  surface-tension  in  a  homogeneous  liquid  film,  and 
was  therefore  equal  in  all  directions,  so  that  T  =  T' ;  and  we  have 
only  dealt  with  surfaces,  or  parts  of  a  surface,  where  extraneous 
pressure,  2^„,  was  non-existent.  Now  in  the  case  of  a  bird's  egg, 
the  external  pressure  2>nj  that  is  to  say  the  pressure  exercised  by 
the  walls  of  the  oviduct,  will  be  found  to  be  a  very  important 
factor ;  but  in  the  case  of  the  yeast-cell  or  the  Spirogyra,  wholly 
immersed  in  water,  no  such  external  pressure  comes  into  play. 
We  are  accordingly  left,  in  such  cases  as  these  last,  with  two 
hypotheses,  namely  that  the  departure  from  a  spherical  form  is  due 
to  inequalities  in  the  internal  pressure  P,  or  else  to  inequalities  in 
the  tension  T,  that  is  to  say  to  a  difference  between  T  and  T' . 
In  other  words,  it  is  theoretically  possible  that  the  oval  form  of 
a  yeast-cell  is  due  to  a  greater  internal  pressure,  a  greater 
"tendency  to  grow,"  in  the  direction  of  the  longer  axis  of  the 
ellipse,  or  alternatively,  that  with  equal  and  symmetrical  tendencies 
to  growth  there  is  associated  a  difference  of  external  resistance  in 

maximum  surface-curvature  lying  at  the  level  where  the  densities  of  the  drop 
and  the  surrounding  liquid  are  just  equal.  The  sectional  outline  of  the  drop  has 
been  shewn  to  be  not  a  true  oval  or  ellipse,  but  a  somewhat  complicated  quartic 
curve.     (Rice,  Phil.  Mag.  Jan.  1915.) 


V]  OF  ASYMMETRY  AND   ANISOTROPY  243 

respect  of  the  tension  of  the  cell-wall.  Now  the  former  hypothesis 
is  not  impossible  ;  the  protoplasm  is  far  from  being  a  perfect  fluid ; 
it  is  the  seat  of  various  internal  forces,  sometimes  manifestly 
polar;  and  accordingly  it  is  quite  possible  that  the  internal 
forces,  osmotic  and  other,  which  lead  to  an  increase  of  the  content 
of  the  cell  and  are  manifested  in  pressure  outwardly  directed 
upon  its  wall  may  be  unsymmetrical,  and  such  as  to  lead  to  a 
deformation  of  what  would  otherwise  be  a  simple  sphere.  But 
while  this  hypothesis  is  not  impossible,  it  is  not  very  easy  of 
acceptance.  The  protoplasm,  though  not  a  perfect  fluid,  has  yet 
on  the  whole  the  properties  of  a  fluid;  within  the  small  compass 
of  the  cell  there  is  little  room  for  the  development  of  unsymmetrical 
pressures ;  and,  in  such  a  case  as  Spirogyra,  where  a  large  part  of 
the  cavity  is  filled  by  a  fluid  and  w^atery  cell-sap,  the  conditions 
are  still  more  obviously  those  under  which  a  uniform  hydrostatic 
pressure  is  to  be  expected.  But  in  variations  of  T,  that  is  to  say 
of  the  specific  surface-tension  per  unit  area,  we  have  an  ample 
field  for  all  the  various  deformations  with  which  we  shall  have  to 
deal.  Our  condition  now  is,  that  [TjR  +  T'/R')  =  a  constant ;  but 
it  no  longer  follows,  though  it  may  still  often  be  the  case,  that  this 
will  represent  a  surface  of  absolute  minimal  area.  As  soon  as  T 
and  T'  become  unequal,  it  is  obvious  that  we  are  no  longer  dealing 
with  a  perfectly  liquid  surface  film ;  but  its  departure  from  a 
perfect  fluidity  may  be  of  all  degrees,  from  that  of  a  slight  non- 
isotropic  viscosity  to  the  state  of  a  firm  elastic  membrane*.  And 
it  matters  little  whether  this  viscosity  or  semi-rigidity  be  mani- 
fested in  the  self-same  layer  which  is  still  a  part  of  the  protoplasm 
of  the  cell,  or  in  a  layer  which  is  completely  differentiated  into  a 
distinct  and  separate  membrane.  As  soon  as,  by  secretion  or 
"adsorption,"  the  molecular  constitution  of  the  surface  layer  is 
altered,  it  is  clearly  conceivable  that  the  alteration,  or  the  secondary 
chemical  changes  which  follow  it,  may  be  such  as  to  produce  an 
anisotropy,  and  to  render  the  molecular  forces  less  capable  in 
one  direction  than  another  of  exerting  that  contractile  force  by 
which  they  are  striving  to  reduce  to  an  absolute  minimum  the 

*  Indeed  any  non-isotropic  stiffness,  even  though  T  remained  uniform,  would 
simulate,  and  be  indistinguishable  from,  a  condition  of  non-stiffness  arid  non- 
isotropic  T. 

16—2 


244  THE  FORMS   OF  CELLS  [ch. 

surface  area  of  the  cell.  A  slight  inequality  in  two  opposite 
directions  will  produce  the  ellipsoid  cell,  and  a  very  great  in- 
equality will  give  rise  to  the  cyHndrical  cell*. 

I  take  it  therefore,  that  the  cylindrical  cell  of  Spirogyra,  or 
any  other  cylindrical  cell  which  grows  in  freedom  from  any 
manifest  external  restraint,  has  assumed  that  particular  form 
simply  by  reason  of  the  molecular  constitution  of  its  developing 
surface-membrane;  and  that  this  molecular  constitution  was 
anisotropous,  in  such  a  way  as  to  render  extension  easier  in  one 
direction  than  another. 

Such  a  lack  of  homogeneity  or  of  isotropy,  in  the  cell-wall  is 
often  rendered  visible,  especially  in  plant-cells,  in  various  ways, 
in  the  form  of  concentric  lamellae,  annular  and  spiral  striations, 
and  the  like. 

But  this  phenomenon,  while  it  brings  about  a  certain  departure 
from  complete  symmetry,  is  still  compatible  with,  and  coexistent 
with,  many  of  the  phenomena  which  we  have  seen  to  be  associated 
with  surface-tension.  The  symmetry  of  tensions  still  leaves  the 
cell  a  solid  of  revolution,  and  its  surface  is  still  a  surface  of  equi- 
librium. The  fluid  pressure  within  the  cylinder  still  causes  the 
film  or  membrane  which  caps  its  ends  to  be  of  a  spherical  form. 
And  in  the  young  cell,  where  the  surface  pellicle  is  absent  or  but 
little  differentiated,  as  for  instance  in  the  oogonium  of  Achlya, 
or  in  the  young  zygospore  of  Spirogyra,  we  always  see  the  tendency 
of  the  entire  structure  towards  a  spherical  form  reasserting  itself : 
unless,  as  in  the  latter  case,  it  be  overcome  by  direct  compression 
within  the  cylindrical  mother-cell.  Moreover,  in  those  cases 
where  the  adult  filament  consists  of  cylindrical  cells,  we  see  that 
the  young,  germinating  spore,  at  first  spherical,  very  soon  assumes 
with  growth  an  elliptical  or  ovoid  form :  the  direct  result  of  an 
incipient  anisotropy  of  its  envelope,  which  when  more  developed 
will  convert  the  ovoid  into  a  cyHnder.  We  may  also  notice  that 
a  truly  cylindrical  cell  is  comparatively  rare;  for  in  most  cases, 
what  we  call  a  cylindrical  cell  shews  a  distinct  bulging  of  its  sides ; 
it  is  not  truly  a  cylinder,  but  a  portion  of  a  spheroid  or  ellipsoid. 

*  A  non-symmetry  of  T  and  T'  might  also  be  capable  of  explanation  as  a  result 
of  "liquid  crystallisation."  This  hypothesis  is  referred  to,  in  connection  with  the 
blood-corpuscles,  on  p.  272. 


vj  OF  ASYMMETRY  AND   ANISOTROPY  245 

Unicellular  organisms  in  general,  including  the  protozoa,  the 
unicellular  cryptogams,  the  various  bacteria,  and  the  free, 
isolated  cells,  spores,  ova,  etc.  of  higher  organisms,  are  referable 
for  the  most  part  to  a  very  small  number  of  typical  forms ;  but 
besides  a  certain  number  of  others  which  may  be  so  referable, 
though  obscurely,  there  are  obviously  many  others  in  which 
either  no  symmetry  is  to  be  recognized,  or  in  which  the  form  is 
clearly  not  one  of  equihbrium.  Among  these  latter  we  have 
Amoeba  itself,  and  all  manner  of  amoeboid  organisms,  and  also 
many  curiously  shaped  cells,  such  as  the  Trypanosomes  and  various 
other  aberrant  Infusoria.  We  shall  return  to  the  consideration  of 
these ;  but  in  the  meanwhile  it  will  suffice  to  say  that,  as  their 
surfaces  are  not  equilibrium-surfaces,  so  neither  are  the  living 
cells  themselves  in  any  stable  equilibrium.  On  the  contrary,  they 
are  in  continual  flux  and  movement,  each  portion  of  the  surface 
constantly  changing  its  form,  and  passing  from  one  phase  to 
another  of  an  equilibrium  which  is  never  stable  for  more  than 
a  moment.  The  former  class,  which  rest  in  stable  equilibrium, 
must  fall  (as  we  have  seen)  into  two  classes, — those  whose  equi- 
librium arises  from  liquid  surface-tension  alone,  and  those  in 
whose  conformation  some  other  pressure  or  restraint  has  been 
superimposed  upon  ordinary  surface-tension. 

To  the  fact  that  these  httle  organisms  belong  to  an  order  of 
magnitude  in  which  form  is  mainly,  if  not  wholly,  conditioned  and 
controlled  by  molecular  forces,  is  due  the  hmited  range  of 
forms  which  they  actually  exhibit.  These  forms  vary  according 
to  varying  physical  conditions.  Sometimes  they  do  so  in  so  regular 
and  orderly  a  way  that  we  instinctively  explain  them  merely  as 
"phases  of  a  life-history/'  and  leave  physical  properties  and 
physical  causation  alone :  but  many  of  their  variations  of  form  we 
treat  as  exceptional,  abnormal,  decadent  or  morbid,  and  are  apt 
to  pass  these  over  in  neglect,  while  we  give  our  attention  to  what 
we  suppose  to  be  the  typical  or  "characteristic"  form  or  attitude. 
In  the  case  of  the  smallest  organisms,  the  bacteria,  micrococci, 
and  so  forth,  the  range  of  form  is  especially  limited,  owing  to  their 
minuteness,  the  powerful  pressure  which  their  highly  curved 
surfaces  exert,  and  the  comparatively  homogeneous  nature  of  their 
substance.     But  within  their  narrow  range  of  possible  diversity 


246 


THE  FORMS  OF  CELLS 


[CH. 


these  minute  organisms  are  protean  in  their  changes  of  form. 
A  certain  species  will  not  only  change  its  shape  from  stage  to 
stage  of  its  Uttle  "  cycle  "  of  life ;  but  it  will  be  remarkably  different 
in  outward  form  according  to  the  circumstances  under  which  we 
find  it,  or  the  histological  treatment  to  which  we  submit  it.  Hence 
the  pathological  student,  commencing  the  study  of  bacteriology, 
is  early  warned  to  pay  little  heed  to  differences  oiform,  for  purposes 
of  recognition  or  specific  identification.  Whatever  grounds  we 
may  have  for  attributing  to  these  organisms  a  permanent  or  stable 
specific  identity  (after  the  fashion  of  the  higher  plants  and  animals), 
we  can  seldom  safely  do  so  on  the  ground  of  definite  and  always 
recognisable  for7n :    we  may  often  be  inclined,  in  short,  to  ascribe 


Fiw.  73.     A  flagellate  "monad,"  Distigma 
proteus,  Ehr.     (After  Saville  Kent.) 


Fig,  74.     Noctiluca  miliaris. 


to  them  a  physiological  (sometimes  a  "pathogenic"),  rather  than 
a  morphological  specificity. 


Among  the  Infusoria,  we  have  a  small  number  of  forms  whose 
symmetry  is  distinctly  spherical,  for  instance  among  the  small 
flagellate  monads ;  but  even  these  are  seldom  actually  spherical 
except  when  we  see  them  in  a  non-flagellate  and  more  or  less 
encysted  or  "resting"  stage.  In  this  condition,  it  need  hardly  be 
remarked  that  the  spherical  form  is  common  and  general  among 
a  great  variety  of  unicellular  organisms.  When  our  little  monad 
developes  a  flagellum,  that  is  in  itself  an  indication  of  "polarity" 
or  symmetrical  non-homogeneity  of  the  cell ;    and  accordingly,  we 


V]  OF  VARIOUS  UNDULOIDS  247 

usually  see  signs  of  an  unequal  tension  of  the  membrane  in  the 
neighbourhood  of  the  base  of  the  fiagellum.  Here  the  tension  is 
usually  less  than  elsewhere,  and  the  radius  of  curvature  is  accord- 
ingly less :  in  other  words  that  end  of  the  cell  is  drawn  out  to  a 
tapering  point  (Fig.  73).  But  sometimes  it  is  the  other  way,  as 
in  Noctiluca,  where  the  large  fiagellum  springs  from  a  depression 
in  the  otherwise  uniformly  rounded  cell.  In  this  case  the  explan- 
ation seems  to  lie  in  the  many  strands  of  radiating  protoplasm 
which  converge  upon  this  point,  and  may  be  supposed  to  keep  it 
relatively  fixed  by  their  viscosity,  while  the  rest  of  the  cell-surface 
is  free  to  expand  (Fig,  74). 

A  very  large  number  of  Infusoria  represent  unduloids,  or 
portions  of  unduloids,  and  this  type  of  surface  appears  and 
reappears  in  a  great  variety  of  forms.  The  cups  of  the  various 
species  of  Vorticella  (Fig.     75)  are  nothing  in  the  world  but  a 


Fig.  75.     Various  species  of  Vorticella.     (Mostly  after  Saville  Kent.) 

beautiful  series  of  unduloids,  or  partial  unduloids,  in  every  grada- 
tion from  a  form  that  is  all  but  cylindrical  to  one  that  is  all  but 
a  perfect  sphere.  These  unduloids  are  not  completely  symmetrical, 
but  they  are  such  unduloids  as  develop  themselves  when  w^e 
suspend  an  oil-globule  between  two  unequal  rings,  or  blow  a 
soap-bubble  betw^een  two  unequal  pipes;  for,  just  as  in  these 
cases,  the  surface  of  our  Vorticella  bell  finds  its  terminal  supports, 
on  the  one  hand  in  its  attachment  to  its  narrow  stalk,  and  on  the 
other  in  the  thickened  ring  from  which  spring  its  circumoral  cilia. 
And  here  let  me  say,  that  a  point  or  zone  from  which  cilia  arise 
would  seem  always  to  have  a  pecuKar  relation  to  the  surrounding 
tensions.  It  usually  forms  a  sharp  salient,  a  prominent  point 
or  ridge,  as  in  our  little  monads  of  Fig.  73;  shewing  that, 
in  its  formation,  the  surface  tension  had  here  locally  diminished. 
But  if  such  a  ridge  or  fillet  consolidate  in  the  least  degree,  it 
becomes  a  source  of  strength,  and  a  'jwint  (Vapfui  for  the  adjacent 
film.     We  shall  deal  with  this  point  again  in  the  next  chapter. 


248 


THE  FORMS  OF  CELLS 


[CH. 


Precisely  the  same  series  of  unduloid  forms  may  be  traced  in 
even  greater  variety  among  various  other  families  or  genera  of  the 


Fig.  76.     Various  species  of  Salpingoeca. 

ft 


Fig.  77.     Various  species  of  Tintinnus,  Dinobryon  and  Codonella.     (After 
Saville  Kent  and  others.) 

Infusoria.  Sometimes,  as  in  Vorticella  itself,  the  unduloid  is  seen 
merely  in  the  contour  of  the  soft  semifluid  body  of  the  living 
animal.  At  other  times,  as  in  Salpingoeca,  Tin- 
tinnus, and  many  other  genera,  we  have  a  distinct 
membranous  cup,  separate  from  the  animal,  but 
originally  secreted  by,  and  moulded  upon,  its 
semifluid  living  surface.  Here  we  have  an  excellent 
illustration  of  the  contrast  between  the  different 
ways  in  which  such  a  structure  may  be  regarded 
and  interpreted.  The  teleological  explanation  is 
that  it  is  developed  for  the  sake  of  protection,  as  a 
domicile  and  shelter  for  the  little  organism  within. 
The  mechanical  explanation  of  the  physicist  (seeking 
only  after  the  "efficient,"  and  not  the  "final"  cause),  is  that  it  is 


Fig.  78. 
Vaginicola. 


V] 


OF  VARIOUS  UNDULOIDS 


249 


FoUiculina. 


present,  and  has  its  actual  conformation,  by  reason  of  certain 
chemico-physical  conditions :  that  it  was  inevitable,  under  the 
given  conditions,  that  certain  constituent 
substances  actually  present  in  the  proto- 
plasm should  be  aggregated  by  molecular 
forces  in  its  surface  layer ;  that  under  this 
adsorptive  process,  the  conditions  con- 
tinuing favourable,  the  particles  should 
accumulate  and  concentrate  till  they 
formed  (with  the  help  of  the  surrounding 
medium)  a  pellicle  or  membrane,  thicker 
or  thinner  as  the  case  might  be ;  that  this 
surface  pellicle  or  membrane  was  inevitably  bound,  by  molecular 
forces,  to  become  a  surface  of  the  least  possible  area  which  the 
circumstances  permitted ;  that  in  the 
present  case,  the  symmetry  and  "  freedom  " 
of  the  system  permitted,  and  ipso  facto 
caused,  this  surface  to  be  a  surface  of 
revolution ;  and  that  of  the  few  surfaces 
of  revolution  which,  as  being  also  surfaces 
?ninimae  areae,  were  available,  the  undu- 
loid  was  manifestly  the  one  permitted, 
and  ipso  facto  caused,  by  the  dimensions 
of  the  organisms  and  other  circumstances 
of  the  case.  And  just  as  the  thickness  or 
thinness  of  the  pellicle  was  obviously  a 
subordinate  matter,  a  mere  matter  of 
degree,  so  we  also  see  that  the  actual 
outline  of  this  or  that  particular  unduloid 
is  also  a  very  subordinate  matter,  such  as 
physico-chemical  variants  of  a  minute  kind 
would  suffice  to  bring  about ;  for  between 
the  various  unduloids  which  the  various 
species  of  Vorticella  represent,  there  is  no 
more  real  difference  than  that  difference 
of  ratio  or  degree  which  exists  between 
two  circles  of  different  diameter,  or  two 
lines  of  unequal  length. 


Fig.  80.     Trachelophyllum. 
(After  Wreszniowski.) 


250  THE   FOKMS   OF  CELLS  [ch. 

In  very  many  cases  (of  which  Fig.  80  is  an  example),  we  have 
an  unduloid  form  exhibited,  not  by  a  surrounding  pelhcle  or  shell, 
but  by  the  soft,  protoplasmic  body  of  a  ciliated  organism.  In 
such  cases  the  form  is  mobile,  and  continually  changes  from  one 
to  another  unduloid  contour,  according  to  the  movements  of  the 
animal.  We  have  here,  apparently,  to  deal  with  an  unstable 
equilibrium,  and  also  sometimes  with  the  more  comphcated 
problem  of  "  stream- lines,"  as  in  the  difficult  problems  suggested 
by  the  form  of  a  fish.  But  this  whole  class  of  cases,  and  of 
problems,  we  can  merely  take  note  of  in  passing,  for  their  treat- 
ment is  too  hard  for  us. 

In  considering  such  series  of  forms  as  the  various  unduloids 
which  we  have  just  been  regarding,  we  are  brought  sharply 
up  (as  in  the  case  of  our  Bacteria  or  Micrococci)  against  the  bio- 
logical concept  of  organic  species.  In  the  intense  classificatory 
activity  of  the  last  hundred  years,  it  has  come  about  that  every 
form  which  is  apparently  characteristic,  that  is  to  say  which  is 
capable  of  being  described  or  portrayed,  and  capable  of  being 
recognised  when  met  with  again,  has  been  recorded  as  a  species, — 
for  we  need  not  concern  ourselves  with  the  occasional  discussions, 
or  individual  opinions,  as  to  whether  such  and  such  a  form  deserve 
"specific  rank,"  or  be  "only  a  variety."  And  this  secular  labour 
is  pursued  in  direct  obedience  to  the  precept  of  the  Systema 
Naturae, — "ut  sic  in  summa  confusione  reriim  apparenti,  siimmvs 
conspiciatur  Naturae  ordo."  In  hke  manner  the  physicist  records, 
and  is  entitled  to  record,  his  many  hundred  "species"  of  snow- 
crystals*,  or  of  crystals  of  calcium  carbonate.  But  regarding 
these  latter  species,  the  physicist  makes  no  assumptions :  he 
records  them  simpliciter,  as  specific  "forms";  he  notes,  as  best 
he  can,  the  circumstances  (such  as  temperature  or  humidity) 
under  which  they  occur,  in  the  hope  of  elucidating  the  conditions 
determining  their  formation ;   but  above  all,  he  does  not  introduce 

*  The  case  of  the  snow-crystals  is  a  particularly  interesting  one;  for  their 
"distribution"  is  in  some  ways  analogous  to  what  we  find,  for  instance,  among  our 
microscopic  skeletons  of  Radiolarians.  That  is  to  say,  we  may  one  day  meet 
with  myriads  of  some  one  particular  form  or  species  only,  and  another  day  with 
myriads  of  another;  while  at  another  time  and  place  we  may  find  species  inter- 
mingled in  inexhaustible  variety.  (Cf.  e.g.  J.  Glaisher,  III.  London  News,  Feb.  17, 
1855;    Q.J. M.S.  iii,  pp.   179-185,  1855). 


V]  OF  FORM   AND   SPECIES  251 

the  element  of  time,  and  of  succession,  or  discuss  their  origin  and 
affihation  as  an  historical  sequence  of  events.  But  in  biology,  the 
term  species  carries  with  it  many  large,  though  often  vague 
assumptions.  Though  the  doctrine  or  concept  of  the  "  permanence 
of  species"  is  dead  and  gone,  yet  a  certain  definite  value,  or  sort 
of  quasi-permanency,  is  still  connoted  by  the  term.  Thus  if  a  tiny 
foraminiferal  shell,  a  Lagena  for  instance,  be  found  living  to-day, 
and  a  shell  indistinguishable  from  it  to  the  eye  be  found  fossil 
in  the  Chalk  or  some  other  remote  geological  formation,  the 
assumption  is  deemed  legitimate  that  that  species  has  "survived," 
and  has  handed  down  its  minute  specific  character  or  characters, 
from  generation  to  generation,  unchanged  for  untold  n\yriads  of 
vears*.  Or  if  the  ancient  forms  be  like  to,  rather  than  identical 
with  the  recent,  we  still  assume  an  unbroken  descent,  accompanied 
by  the  hereditary  transmission  of  common  characters  and  pro- 
gressive variations.  And  if  two  identical  forms  be  discovered  at 
the  ends  of  the  earth,  still  (with  occasional  slight  reservations  on 
the  score  of  possible  "homoplasy"),  we  build  hypotheses  on  this 
fact  of  identity,  taking  it  for  granted  that  the  two  appertain  to 
a  common  stock,  whose  dispersal  in  space  must  somehow  be 
accounted  for,  its  route  traced,  its  epoch  determined,  and  its 
causes  discussed  or  discovered.  In  short,  the  naturalist  admits 
no  exception  to  the  rule  that  a  "natural  classification"  can  only 
be  a  genealogical  one,  nor  ever  doubts  that  "  The  fact  that  we  are 
able  to  classify  organisms  at  all  in  accordance  with  the  structural 
characteristics  ivhich  they  fresent,  is  due  to  the  fact  of  their  being 
related  by  descenti"  But  this  great  generahsation  is  apt  in  my 
opinion,  to  carry  us  too  far.  It  may  be  safe  and  sure  and  helpful 
and  illuminating  when  we  apply  it  to  such  complex  entities, — 
such  thousand-fold  resultants  of  the  combination  and  permutation 
of  many  variable  characters,^ — as  a  horse,  a  lion  or  an  eagle ; 
but  (to  my  mind)  it  has  a  very  different  look,  and  a  far  less  firm 
foundation,  when  we  attempt  to  extend  it  to  minute  organisms 
whose  specific  characters  are  few  and  simple,  whose  simplicity 

*  Cf.  Bergson,  Creative  Evolution,  p.  107:  "Certain  Foraminifera  have  not 
varied  since  the  Sihirian  epoch.  Unmoved  witnesses  of  the  innumerable  revolu- 
tions that  have  upheaved  our  planet,  the  Lingulae  are  today  what  they  were  at 
the  remotest  times  of  the  palaeozoic  era." 

t  Ray  Lankester,  A.M.N.H.  (4),  xi,  p.  .321,  1873. 


252  THE   FORMS   OF   CELLS  [ch. 

becomes  much  more  manifest  when  we  regard  it  from  the  point 
of  view  of  physical  and  mathematical  description  and  analysis, 
and  whose  form  is  referable,  or  (to  say  the  least  of  it)  is  very 
largely  referable,  to  the  direct  and  immediate  action  of  a  particular 
physical  force.  When  we  come  to  deal  with  the  minute  skeletons 
of  the  Radiolaria  we  shall  again  find  ourselves  dealing  with  endless 
modifications  of  form,  in  which  it  becomes  still  more  di£6-cult  to 
discern,  or  to  apply,  the  guiding  principle  of  affiliation  oi  genealogy. 
Among  the  more  aberrant  forms  of  Infusoria  is  a  little  species 
known  as  Trichodina  fediculus,  a  parasite  on  the  Hydra,  or  fresh- 
water polype  (Fig.  8L )   This  Trichodina  has  the  form  of  a  more  or  less 

flattened  circular  disc,  with  a  ring 
of  cilia  around  both  its  upper  and 
lower  margins.  The  salient  ridge 
from  which  these  cilia  spring  may 
be  taken,  as  we  have  already  said, 
to  play  the  part  of  a  strengthening 
"fillet."  The  circular  base  of  the 
animal  is  flattened,  in  contact  with 
the  flattened  surface  of  the  Hydra 
over  which  it  creeps,  and  the  oppo- 
site, upper  surface  may  be  flattened  nearly  to  a  plane,  or  may  at 
other  times  appear  slightly  convex  or  slightly  concave.  The  sides 
of  the  little  organism  are  contracted,  forming  a  symmetrical 
equatorial  groove  between  the  upper  and  lower  discs ;  and,  on 
account  of  the  minute  size  of  the  animal  and  its  constant 
movements,  we  cannot  submit  the  curvature  of  this  concavity  to 
measurement,  nor  recognise  by  the  eye  its  exact  contour.  But 
it  is  evident  that  the  conditions  are  precisely  similar  to  those 
described  on  p.  223,  where  we  were  considering  the  conditions 
of  stability  of  the  catenoid.  And  it  is  further  evident  that,  when 
the  upper  disc  is  actually  plane,  the  equatorial  groove  is  strictly 
a  catenoid  surface  of  revolution ;  and  when  on  the  other  hand  it 
is  depressed,  then  the  equatorial  groove  will  tend  to  assume 
the  form  of  a  nodoidal  surface. 

Another  curious  type  is  the  flattened  spiral  of  Dinenympha^ 

*  Leidy,  Parasites  of  the  Termites,  J.  Nat.  Sci.,  Philadelphia,  vm,  pp.  425- 
447,  1874—81;    cf.  Saville  Kent's  Infusoria,  ii,  p.  551. 


V] 


OF   COLLAR-CELLS 


253 


which  reminds  us  of  the  cylindrical  spiral  of  a  Spirillum  among 
the  bacteria.  In  Dinenympha  we  have  a  symmetrical  figm-e,  whose 
two  opposite  surfaces  each  constitute  a  surface  of  constant  mean 
curvature ;  it  is  evidently  a  figure  of  equilibrium  under  certain 
special  conditions  of  restraint.  The  cylindrical  coil  of  the 
Spirillum,  on  the  other  hand,  is  a  surface  of  constant  mean  curva- 
ture, and  therefore  of  equilibrium,  as  truly,  and  in  the  same  sense, 
as  the  cylinder  itself. 


\ 

Fig.  82.     Dinenympha  gracilis,  Leidy. 

A  very  curious  conformation  is  that  of  the  vibratile  "collar," 
found  in  Codosiga  and  the  other  "Choanoflagellates,"  and  which 
we  also  meet  with  in  the  "collar-cells"  which  line  the  interior 
cavities  of  a  sponge.  Such  collar-cells  are  always  very  minute, 
and  the  collar  is  constituted  of  a  very  delicate  film,  which 
shews  an  undulatory  or  ripphng  motion.  It  is  a  surface  of 
revolution,  and  as  it  maintains  itself  in  equilibrium  (though  a 
somewhat  unstable  and  fluctuating  one),  it  must  be,  under  the 
restricted  circumstances  of  its  case,  a  surface  of  minimal  area. 
But  it  is  not  so  easy  to  see  what  these  special  circumstances  are ; 
and  it  is  obvious  that  the  collar,  if  left  to  itself,  must  at  once 


254  THE   FORMS   OF  CELLS  [ch. 

contract  downwards  towards  its  base,  and  become  confluent  with 
the  general  surface  of  the  cell ;  for  it  has  no 
longitudinal  supports  and  no  strengthening  ring 
at  its  periphery.  But  in  all  these  collar-cells, 
there  stands  within  the  annulus  of  the  collar 
a  large  and  powerful  cilium  or  flagellum,  in 
constant  movement;  and  by  the  action  of  this 
flagellum,  and  doubtless  in  part  also  by  the 
intrinsic  vibrations  of  the  collar  itself,  there  is 
set  up  a  constant  steady  current  in  the  sur- 
rounding water,  whose  direction  would  seem  to 
be  such  that  it  passes  up  the  outside  of  the 
collar,  down  its  inner  side,  and  out  in  the  middle 
in  the  direction  of  the  flagellum ;  and  there  is  a 
Fig.  83.^  distinct  eddy,  in  which  foreign  particles  tend  to 

be  caught,  around  the  peripheral  margin  of  the  collar.  When  the 
cell  dies,  that  is  to  say  when  motion  ceases,  the  collar  immediately 
shrivels  away  and  disappears.  It  is  notable,  by  the  way,  that 
the  edge  of  this  little  mobile  cup  is  always  smooth,  never  notched 
or  lobed  as  in  the  cases  we  have  discussed  on  p.  236:  this  latter 
condition  being  the  outcome  of  a  definite  instability,  marking  the 
close  of  a  period  of  equilibrium;  while  in  the  vibratile  collar  of 
Codosiga  the  equilibrium,  such  as  it  is,  is  being  constantly 
renewed  and  perpetuated  like  that  of  a  juggler's  pole,  by  the 
motions  of  the  system.  I  .take  it  that,  somehow,  its  existence 
(in  a  state  of  partial  equilibrium)  is  due  to  the  current  motions, 
and  to  the  traction  exerted  upon  it  through  the  friction  of 
the  stream  which  is  constantly  passing  by.  I  think,  in  short, 
that  it  is  formed  very  much  in  the  same  way  as  the  cup-like  ring 
of  streaming  ribbons,  which  we  see  fluttering  and  vibrating  in  the 
air-current  of  a  ventilating  fan. 

It  is  likely  enough,  however,  that  a  different  and  much  better 
explanation  may  yet  be  found;  and  if  we  turn  once  more  to 
Mr  Worthington's  Study  of  Splashes,  we  may  find  a  curious 
suggestion  of  analogy  in  the  beautiful  craters  encircling  a  central 
jet  (as  the  collar  of  Codosiga  encircles  the  flagellum),  which  we  see 
produced  in  the  later  stages  of  the  splash  of  a  pebble*. 

*  Op.  cit.  p.  79. 


i 


v]  OF   THE   SIMPLER  FORAMINIFEHA  255 

Among  the  Foraminifera  we  have  an  immense  variety  of  forms, 
Avhich,  in  the  hght  of  surface  tension  and  of  the  principle  of 
minimal  area,  are  capable  of  explanation  and  of  reduction  to  a 
small  number  of  characteristic  types.  Many  of  the  Foraminifera 
are  composite  structures,  formed  by  the  successive  imposition  of 
cell  upon  cell,  and  these  we  shall  deal  with  later  on ;  let  us  glance 
here  at  the  simpler  conformations  exhibited  by  the  single  cham- 
bered or  "  monothalamic "  genera,  and  perhaps  one  or  two  of  the 
simplest  composites. 

We  begin  with  forms,  like  Astrorhiza  (Fig.  219,  p.  464),  which 
are  in  a  high  degree  irregular,  and  end  with  others  which  manifest  a 
perfect  and  mathematical  regularity.  The  broad  difference  between 
these  two  types  is  that  the  former  are  characterised,  like  Amoeba, 
by  a  variable  surface  tension,  and  consequently  by  unstable  equi- 
librium ;  but  the  strong  contrast  between  these  and  the  regular  forms 
is  bridged  over  by  various  transition-stages,  or  differences  of  degree. 
Indeed,  as  in  all  other  Rhizopods,  the  very  fact  of  the  emission  of 
pseudopodia,  which  reach  their  highest  development  in  this  group 
of  animals,  is  a  sign  of  unstable  surface-equilibrium ;  and  we  must 
therefore  consider  that  those  forms  which  indicate  symmetry  and 
equilibrium  in  their  shells  have  secreted  these  during  periods  when 
rest  and  uniformity  of  surface  conditions  alternated  with  the 
phases  of  pseudopodial  activity.  The  irregular  forms  are  in 
almost  all  cases  arenaceous,  that  is  to  say  they  have  no  sohd  shells 
formed  by  steady  adsorptive  secretion,  but  only  a  looser  covering 
of  sand  grains  with  which  the  protoplasmic  body  has  come  in 
contact  and  cohered.  Sometimes,  as  in  Ramulina,  we  have  a 
calcareous  shell  combined  with  irregularity  of  form ;  but  here  we 
can  easily  see  a  partial  and  as  it  were  a  broken  regularity,  the 
regular  forms  of  sphere  and  cylinder  being  repeated  in  various 
parts  of  the  ramified  mass.  When  Ave  look  more  closely  at  the 
arenaceous  forms,  we  find  that  the  same  thing  is  true  of  them ; 
they  represent,  either  in  whole  or  part,  approximations  to  the  form 
of  surfaces  of  equilibrium,  spheres,  cylinders  and  so  forth.  In 
Aschemonella  w^e  have  a  precise  replica  of  the  calcareous  Ramulina  ; 
and  in  Astrorhiza  itself,  in  the  forms  distinguished  by  naturalists 
as  A.  crassatina,  what  is  described  as  the  " subsegmented  interior*  " 
*  Brady,  Challenger  Monograph,  pi.  xx,  p.  233. 


256 


THE  FOKMS   OF  CELLS 


[CH. 


seems  to  shew  the  natural,  physical  tendency  of  the  long  semifluid 
cylinder  of  protoplasm  to  contract,  at  its  limit  of  stability,  into 
unduloid  constrictions,  as  a  step  towards  the  breaking  up  into 
separate  spheres :  the  completion  of  which  process  is  restrained  or 
prevented  by  the  rigidity  and  friction  of  the  arenaceous  covering. 
Passing  to  the  typical,  calcareous-shelled  Foraminifera,  we  have 
the  most  symmetrical  of  all  possible  types  in  the  perfect  sphere  of 
Orbulina ;  this  is  a  pelagic  organism,  whose  floating  habitat  places 


Fig.  84.     Various  species  of  Lagena.     (After  Brady.) 


it  in  a  position  of  perfect  symmetry  towards  all  external  forces. 
Save  for  one  or  two  other  forms  which  are  also  spherical,  or 
approximately  so,  like  Thurammina,  the  rest  of  the  monothalamic 
calcareous  Foraminifera  are  all  comprised  by  naturalists  within 
the  genus  Lagena.  This  large  and  varied  genus  consists  of  "flask- 
shaped"  shells,  whose  surface  is  simply  that  of  an  unduloid,  or 
more  frequently,  like  that  of  a  flask  itself,  an  unduloid  combined 
with  a  portion  of  a  sphere.     We  do  not  know  the  circumstances 


OF  HANGING  DROPS 


257 


under  which  the  shell  of  Lagena  is  formed,  nor  the  nature  of  the 
force  by  which,  during  its  formation,  the  surface  is  stretched  out 
into  the  unduloid  form ;  but  we  may  be  pretty  sure  that  it  is 
suspended  vertically  in  the  sea,  that  is  to  say  in  a  position  of 
symmetry  as  regards  its  vertical  axis,  about  which  the  unduloid 
surface  of  revolution  is  symmetrically  formed.  At  the  same  time 
we  have  other  types  of  the  same  shell  in  which  the  form  is  more 
or  less  flattened ;  and  these  are  doubtless  the  cases  in  which  such 
symmetry  of  position  was  not  present,  or  was  replaced  by  a  broader, 
lateral  contact  with  the  surface  pellicle*. 

While  Orbulina  is  a  simple  spherical  drop,  Lagena  suggests  to 
our  minds  a  "hanging  drop,"  drawn  out  to  a  long  and  slender 
neck  by  its  own  weight,  aided  by  the  viscosity  of  the  material. 


Fig.  8.5.     (After  Darling. 


Indeed  the  various  hanging  drops,  such  as  Mr  C.  R.  Darling  shews 
us,  are  the  most  beautiful  and  perfect  unduloids,  with  spherical 
ends,  that  it  is  possible  to  conceive.  A  suitable  liquid,  a  little 
denser  than  water  and  incapable  of  mixing  with  it  (such  as 
ethyl    benzoate),   is    poured   on   a  surface  of  water.     It  spreads 

*  That  the  Foraminifera  not  only  can  but  do  hang  from  the  surface  of  the 
water  is  confirmed  by  the  following  apt  quotation  which  I  owe  to  Mr  E.  Heron- 
Allen:  "Quand  on  place,  comme  il  a  ete  dit,  le  depot  provenant  du  lavage  des 
fucus  dans  un  flacon  que  Ton  remplit  de  nouveUe  eau,  on  voit  au  bout  d'une  heure 
environ  les  animaux  [Gramid  dujardinii]  se  mettre  en  mouvement  et  commencer 
a  grimper.  Six  heures  apres  ils  tapissent  I'exterieur  du  flacon,  de  sorte  que  les  plus 
eleves  sont  a  trenjte-six  ou  quarante-deux  millimetres  du  fond;  le  lendemain 
beaucoup  d'entre  eux,  apres  avoir  atteint  le  niveau  du  liquide,  ont  continue  a  ramper 
'a  sa  surface,  en  se  laissani  pendre  au-dessous  comme  certains  mollusques  gastero- 
podes."  (Dujardin,  F.,  Observations  nouvelles  sur  les  pretendus  cephalopodes 
microscopiques,  Ann.  des  Sci.  Nat.  (2),  iii,  p.  312,  1835.) 

T.  n.  17 


258  THE   FORMS   OF  CELLS  [ch. 

over  the  surface  and  gradually  forms  a  hanging  drop,  approxi- 
mately hemispherical ;  but  as  more  liquid  is  added  the  drop 
sinks  or  rattier  grows  downwards,  still  adhering  to  the  surface 
film ;  and  the  balance  of  forces  between  gravity  and  surface 
tension  results  in  the  unduloid  contour,  as  the  increasing  weight 
of  the  drop  tends  to  stretch  it  out  and  finally  break  it  in  two. 
At  the  moment  of  rupture,  by  the  way,  a  tiny  droplet  is  formed 
in  the  attenuated  neck,  such  as  we  described  in  the  normal 
division  of  a  cylindrical  thread  (p.  233). 

To  pass  to  a  much  more  highly  organised  class  of  animals,  we  find  the 
unduloid  beautifully  exemplified  in  the  little  flask-shaped  shells  of  certain 
Pteropod  mollusca,  e.g.  Cuvierina*.  Here  again  the  symmetry  of  the  figure 
would  at  once  lead  us  to  suspect  that  the  creature  lived  in  a  position  of 
symmetry  to  the  surrounding  forces,  as  for  instance  if  it  floated  in  the  ocean 
in  an  erect  position,  that  is  to  say  with  its  long  axis  coincident  with  the  direction 
of  gravity;  and  this  we  know  to  be  actually  the  mode  of  life  of  the  little 
Pteropod. 

Many  species  of  Lagena  are  complicated  and  beautified  by  a 
pattern,  and  some  by  the  superadd! tion  to  the  shell  of  plane 
extensions  or  "wings."  These  latter  give  a  secondary,  bilateral 
symmetry  to  the  little  shell,  and  are  strongly  suggestive  of  a 
phase  or  period  of  growth  in  which  it  lay  horizontally  on  the 
surface,  instead  of  hanging  vertically  from  the  surface-film :  in 
which,  that  is  to  say,  it  was  a  floating  and  not  a  hanging 
drop.  The  pattern  is  of  two  kinds.  Sometimes  it  consists 
of  a  sort  of  fine  reticulation,  with  rounded  or  more  or 
less  hexagonal  interspaces :  in  other  cases  it  is  produced  by  a 
symmetrical  series  of  ridges  or  folds,  usually  longitudinal,  on  the 
body  of  the  flask-shaped  cell,  but  occasionally  transversely  arranged 
upon  the  narrow  neck.  The  reticulated  and  folded  patterns  we 
may  consider  separately.  The  netted  pattern  is  very  similar  to  the 
wrinkled  surface  of  a  dried  pea,  or  to  the  more  regular  wrinkled 
patterns  upon  many  other  seeds  and  even  pollen-grains.  If  a 
spherical  body  after  developing  a  "skin"  begin  to  shrink  a  little, 
and  if  the  skin  have  so  far  lost  its  elasticity  as  to  be  unable  to 
keep  pace  with  the  shrinkage  of  the  inner  mass,  it  will  tend  to 
fold  or  wrinkle ;  and  if  the  shrinkage  be  uniform,  and  the  elasticity 
.and  flexibility  of  the  skin  be  also  uniform,  then  the  amount  of 
*  Cf.  Boas,  Spolia  Atlantica,  1886,  pi.  6. 


V]  OF  RETICULATE   OR  WRINKLED   CELLS         259 

folding  will  be  uniformly  distributed  over  the  surface.  Little 
concave  depressions  will  appear,  regularly  interspaced,  and 
separated  by  convex  folds.  The  little  concavities  being  of  equal 
size  (unless  the  system  be  otherwise  perturbed)  each  one  will  tend 
to  be  surrounded  by  six  others  ;  and  when  the  process  has  reached 
its  hmit,  the  intermediate  boundary-walls,  or  raised  folds,  will  be 
found  converted  into  a  regular  pattern  of  hexagons. 

But  the  analogy  of  the  mechanical  wrinkling  of  the  coat  of 
a  seed  is  but  a  rough  and  distant  one ;  for  we  are  evidently  dealing 
with  molecular  rather  than  with  mechanical  forces.  In  one  of 
Darlmg's  experiments,  a  little  heavy  tar-oil  is  dropped  onto  a 
saucer  of  water,  over  which  it  spreads  in  a  thin  film  showing 
beautiful  interference  colours  after  the  fashion  of  those  of  a  soap- 
bubble.  Presently  tiny  holes  appear  in  the  film,  which  gradually 
increase  in  size  till  they  form  a  cellular  pattern  or  honeycomb, 
the  oil  gathering  together  in  the  meshes  or  walls  of  the  cellular 
net.  Some  action  of  this  sort  is  in  all  probability  at  work  in  a 
surface-film,  of  protoplasm  covering  the  shell.  As  a  physical 
phenomenon  the  actions  involved  are  by  no  means  fully  under- 
stood, but  surface-tension,  diffusion  and  cohesion  doubtless  play 
their  respective  part?  therein*.  The  very  perfect  cellular  patterns 
obtained  by  Leduc  (to  which  we  shall  have  occasion  to  refer  in 
a  subsequent  chapter)  are  diffusion  patterns  on  a  larger  scale,  but 
not  essentially  different. 

The  folded  or  pleated  pattern  is  doubtless  to  be  explained,  in 
a  general  way,  by  the  shrinkage  of  a  surface-film  under  certain 

*  This  cellular  pattern  would  seem  to  be  related  to  the  "cohesion  figures" 
described  by  Tomlinson  in  various  surface-films  (Phil.  Mag.  1861  to  1870) ;  to 
the  "tesselated  structure"  in  hquids  described  by  Professor  James  Thomson  in 
1882  [Collected  Papers,  p.  136);  and  to  the  tourbiUo7is  cellulaves  of  Prof.  H.  Benarcl 
(Ann.  de  Chimie  (7),  xxni,  pp.  62-144,  1901,  (8),  xxiv,  pp.  563-566,  1911), 
Rev.  geae'r.  des  Sci.  xi,  p.  1268,  1900;  cf.  also  E.  H.  Weber.  Prggend.  Ann. 
xciv,  p.  452,  1855,  etc.).  The  phenomenon  is  of  great  interest  and  various 
appearances  have  been  referred  to  it,  in  biology,  geology,  metallurgy  and  even 
astronomy :  for  the  flocculent  clouds  in  the  solar  photosphere  shew  an  analogous 
configuration.  (See  letters  by  Kerr  Cirant,  Larmor,  Wager  and  others,  in  Nature, 
April  16  to  June  11,  1914.)  In  many  instances,  marked  by  strict  symmetry  or 
regularity,  it  is  very  possible  that  the  interference  of  waves  or  ripples  may  play 
its  part  in  the  phenomenon.  But  in  the  majority  of  cases,  it  is  fairly  certain  that 
locahsed  centres  of  action,  or  of  diminished  tension,  are  present,  such  as  might  be 
provided  by  dust-particles  in  the  case  of  Darling's  experiment  (cf.  infra,  p.  590). 

17—2 


260  THE   FORMS   OF   CELLS  [ch. 

conditions  of  viscous  or  frictional  restraint.  A  case  which  (as  it 
seems  to  me)  is  closely  analogous  to  that  of  our  foraminiferal 
shells  is  described  by  Quincke*,  who  let  a  film  of  albumin  or  of 
resin  set  and  harden  upon  a  surface  of  quicksilver,  and  found 
that  the  little  solid  pelhcle  had  been 
thrown  into  a  pattern  of  symmetrical 
folds.  If  the  surface  thus  thrown  into 
folds  be  that  of  a  cylinder,  or  any  other 
figure  with  one  principal  axis  of  sym- 
metry, such  as  an  ellipsoid  or  unduloid, 
the  direction  of  the  folds  will  tend  to 
be  related  to  the  axis  of  symmetry, 
^^  and     we     might    expect    accordingly    to 

find  regular  longitudinal,  or  regular  transverse  wrinkling.  Now 
as  a  matter  of  fact  we  almost  invariably  find  in  the  Lagena 
the  former  condition :  that  is  to  say,  in  our  ellipsoid  or  unduloid 
cell,  the  puckering  takes  the  form  of  the  vertical  fluting  on 
a  column,  rather  than  that  of  the  transverse  pleating  of  an 
accordion.  And  further,  there  is  often  a  tendency  for  such 
longitudinal  flutings  to  be  more  or  less  localised  at  the  end  of  the 
ellipsoid,  or  in  the  region  where  the  unduloid  merges  into  its 
spherical  base.  In  this  latter  region  we  often  meet  with  a  regular 
series  of  short  longitudinal  folds,  as  we  do  in  the  forms  of  Lagena 
denominated  L.  semistrinta.  All  these  various  forms  of  surface 
can  be  imitated,  or  rather  can  be  precisely  reproduced,  by  the  art 
of  the  glass-blower  f. 

Furthermore,  they  remind  one,  in  a  striking  way,  of  the 
regular  ribs  or  flutings  in  the  film  or  sheath  which  splashes  up  to 
envelop  a  ^smooth  ball  which  has  been  dropped  into  a  liquid,  as 
Mr  W(  r  hington  has  so  beauiifully  shewn  J. 

*  Ueber  piiysikalischen  Eigenschaften  dunner,  fester  Lamellen,  S.B.  Berlin. 
Akad.   1888,  pp.  789,  790. 

t  Certain  palaeontologists  (e.g.  Haeusler  and  Spandel)  have  maintained  that 
in  each  family  or  genus  the  plain  smooth-shelled  forms  are  the  primitive  and  ancient 
ones,  and  that  the  ribbed  and  otherwise  ornamented  shells  make  their  appearance 
at  later  dates  in  the  course  of  a  definite  evolution  (cf.  Rhumbler,  Foramini/eren 
der  Pla:'kton-Exi:)edition,  1911,  i,  p.  21).  If  this  were  true  it  would  be  of  funda- 
mental importance :    but  this  book  of  mine  would  not  deserve  to  be  written. 

X  A  Study  of  Splashes,  p.  116. 


V]        OF  FLUTED  OR  PLEATED  CELLS       261 

In  Mr  Worthington's  experiment,  there  appears  to  be  something 
of  the  nature  of  a  viscous  drag  in  the  surface-pellicle ;  but  whatever 
be  the  actual  cause  of  variation  of  tension,  it  is  not  difficult  to 
see  that  there  must  be  in  general  a  tendency  towards  longitudinal 
puckering  or  "fluting"  in  the  case  of  a  thin- walled  cylindrical  or 
other  elongated  body,  rather  than  a  tendency  towards  transverse 
puckering,  or  "pleating."  For  let  us  suppose  that  some  change 
takes  place  involving  an  increase  of  surface-tension  in  some  small 
area  of  the  curved  wall,  and  leading  therefore  to  an  increase  of 
pressure :  that  is  to  say  let  T  become  T  +  t,  and  P  become  P  +  f. 
Our  new  equation  of  equilibrium,  then,  in  place  of  P  =  Tjr  +  T/r' 
becomes 


P^         T+tT+t 

P  +  P  = \ 7— 


and  by  subtraction, 


])  =  t/r  +  t/r'. 

Now  if  r  <  r',         t/r  >  t/r'. 

Therefore,  in  order  to  produce  the  small  increment  of  pressure  f, 
it  is  easier  to  do  so  by  increasing  t/r  than  t/r' ;  that  is  to  say,  the 
easier  way  is  to  alter,  or  diminish  r.  And  the  same  will  hold  good 
if  the  tension  and  pressure  be  diminished  instead  of  increased. 

This  is  as  much  as  to  say  that,  when  corrugation  or  "rippling" 
of  the  walls  takes  place  owing  to  small  changes  of  surface-tension, 
and  consequently  of  pressure,  such  corrugation  is  more  likely  to 
take  place  in  the  plane  of  r, — that  is  to  say,  in  the  plane  of  greatest 
curvature.  And  it  follows  that  in  such  a  figure  as  an  ellipsoid, 
wrinkling  will  be  most  likely  to  take  place  not  only  in  a  longitudinal 
direction  but  near  the  extremities  of  the  figure,  that  is  to  say  again 
in  the  region  of  greatest  curvature. 

The  longitudinal  wrinkhng  of  the  flask-shaped  bodies  of  our 
Lagenae,  and  of  the  more  or  less  cylindrical  cells  of  many  other 
Foraminifera  (Fig.  87),  is  in  complete  accord  with  the  above  theo- 
retical considerations  ;  but  nevertheless,  we  soon  find  that  our  result 
is  not  a  general  one,  but  is  defined  by  certain  limiting  conditions, 
and  is  accordingly  subject  to  what  are,  at  first  sight,  important 
exceptions.  For  instance,  when  we  turn  to  the  narrow  neck  of 
the  Lagena  we  see  at  once  that  our  theory  no  longer  holds ;    for 


262 


THE   FORMS   OF   CELLS 


[CH. 


the  wrinkling  which  was  invariably  longitudinal  in  the  body  of 
the  cell  is  as  invariably  transverse  in  the  narrow  neck.  The  reason 
for  the  difference  is  not  far  to  seek.  The  conditions  in  the  neck 
are  very  different  from  those  in  the  expanded  portion  of  the  cell : 
the  main  difference  being  that  the  thickness  of  the  wall  is  no  longer 
insignificant,  but  is  of  considerable  magnitude  as  compared  with 
the  diameter,  or  circumference,  of  the  neck.  We  must  accordingly 
take  it  into  account  in  considering  the  bending  ynoments  at  any 
point  in  this  region  of  the  shell-wall.  And  it  is  at  once  obvious 
that,  in  any  portion  of  the  narrow  neck,  fl,exnre  of  a  wall  in  a 


Fig.  87.     Nodosaria  scalaris, 
Batsch. 


Fig.  88.  Gonangia  of  Campanularians. 
(a)  C.  gracilis;  (b)  C.  grandis. 
(After  Allman.) 


transverse  direction  will  be  very  difficult,  while  flexure  in  a 
longitudinal  direction  will  be  comparatively  easy;  just  as,  in  the 
case  of  a  long  narrow  strip  of  iron,  we  may  easily  bend  it  into 
folds  running  transversely  to  its  long  axis,  but  not  the  other  way. 
The  manner  in  which  our  little  Lagena-shell  tends  to  fold  or  wrinkle, 
longitudinally  in  its  wider  part,  and  transversely  or  annularly  in 
its  narrow  neck,  is  thus  completely  and  easily  explained. 

An  identical  phenomenon  is  apt  to  occur  in  the  little  flask- 
shaped  gonangia,  or  reproductive  capsules,  of  some  of  the  hydroid 
zoophytes.  In  the  annexed  drawings  of  these  gonangia  in  two 
species  of  Campanularia,  we  see  that  in  one  case  the  little  vesicle 


V] 


OF  THE   SIMPLER   FORAMINIFERA 


263 


has  the  flask-shaped  or  unduloid  configuration  of  a  Lagena ;  and 
here  the  walls  of  the  flask  are  longitudinally  fluted,  just  after  the 
manner  we  have  witnessed  in  the  latter  genus.  But  in  the  other 
Campanularian  the  vesicles  are  long,  narrow  and  tubular,  and  here 
a  transverse  folding  or  pleating  takes  the  place  of  the  longitudinally 
fluted  pattern.  And  the  very  form  of  the  folds  or  pleats  is 
enough  to  suggest  that  we  are  not  dealing  here  with  a  simple 
phenomenon  of  surface-tension,  but  with  a  condition  in  which 
surface-tension  and  stiffness  are  both  present,  and  play  their 
parts  in  the  resultant  form. 

Passing  from  the  solitary  flask-shaped  cell  of  Lagena,  we  have, 
in  another  series  of  forms,  a  constricted  cylinder,  or  succession 


a  h  c  d  e  f  g 

Fig.  89.  Various  Foraminifera  (after  Brady),  a,  Nodosaria  simplex;  b,  N. 
pygmaea;  c,  N.  costulata;  e,  N.  hispida;  f,  N.  elata;  d,  Rheophax  (Lituola) 
distans;   g,  Sagrina  virgata. 

of  unduloids ;  such  as  are  represented  in  Fig.  89,  illustrating 
certain  species  of  Nodosaria,  Rheophax  and  Sagrina.  In  some  of 
these  cases,  and  certainly  in  that  of  the  arenaceous  genus  Rheophax, 
we  have  to  do  with  the  ordinary  phenomenon  of  a  segmenting  or 
partially  segmenting  cylinder.  But  in  others,  the  structure  is 
not  developed  out  of  a  continuous  protoplasmic  cylinder,  but  as 
we  can  see  by  examining  the  interior  of  the  shell,  it  has  been 
formed  in  successive  stages,  beginning  with  a  simple  unduloid 
"Lagena,"  about  whose  neck,  after  its  solidification,  another  drop 
of  protoplasm  accumulated,  and  in  turn  assumed  the  unduloid, 
or  lagenoid,  form.     The  chains  of  interconnected  bubbles  which 


264  THE   FORMS   OF   CELLS  [ch. 

Morey  and  Draper  made  many  years  ago  of  melted  resin  are  a 
very  similar  if  not  identical  phenomenon*. 

There  now  remain  for  our  consideration,  among  the  Protozoa, 
the  great  oceanic  group  of  the  Radiolaria,  and  the  little  group  of 
their  freshwater  allies,  the  Heliozoa.  In  nearly  all  these  forms  we 
have  this  specific  chemical  difference  from  the  Foraminifera,  that 
when  they  secrete,  as  they  generally  do  secrete,  a  hard  skeleton, 
it  is  composed  of  silica  instead  of  lime.  These  organisms  and  the 
various  beautiful  and  highly  complicated  skeletal  fabrics  which 
they  develop  give  us  many  interesting  illustrations  of  physical 
phenomena,  among  which  the  manifestations  of  surface-tension 
are  very  prominent.  But  the  chief  phenomena  connected  with 
their  skeletons  we  shall  deal  with  in  another  place,  under  the  head 
of  spicular  concretions. 

In  a  simple  and  typical  Heliozoan,  such  as  the  Sun-animalcule, 
Actinophrys  sol,  we  have  a  "drop"  of  protoplasm,  contracted  by 
its  surface  tension  into  a  spherical  form.  Within  the  heterogeneous 
protoplasmic  mass  are  more  fluid  portions,  and  at  the  surface 
which  separates  these  from  the  surrounding  protoplasm  a  similar 
surface  tension  causes  them  also  to  assume  the  form  of  spherical 
"vacuoles,"  which  in  reality  are  little  clear  drops  within  the  big 
one ;  unless  indeed  they  become  numerous  and  closely  packed,  in 
which  case,  instead  of  isolated  spheres  or  droplets  they  will 
constitute  a  "froth,"  their  mutual  pressures  and  tensions  giving 
rise  to  regular  configurations  such  as  we  shall  study  in  the  next 
chapter.  One  or  more  of  such  clear  spaces  may  be  what  js  called 
a  "contractile  vacuole":  that  is  to  say,  a  droplet  whose  surface 
tension  is  in  unstable  equilibrium  and  is  apt  to  vanish  altogether, 
so  that  the  definite  outline  of  the  vacuole  suddenly  disappears  "j". 
Again,  within  the  protoplasm  are  one  or  more  nuclei,  whose  own 
surface  tension  (at  the  surface  between  the  nucleus  and  the 
surrounding  protoplasm),  has  drawn  them  in  turn  into  the  shape 

*  See  SiUi7na7i''s  Journal,  n,  p.  179,  1820;  and  cf.  Plateau,  op.  cit.  ii,  pp.  134, 
461. 

t  The  presence  or  absence  of  the  contractile  vacuole  or  vacuoles  is  one  of  the 
chief  distinctions,  in  systematic  zoology,  between  the  Heliozoa  and  the  Radiolaria. 
As  we  have  seen  on  p.  165  (footnote),  it  is  probably  no  more  than  a  physical  con- 
^sequence  of  the  different  conditions  of  existence  in  fresh  water  and  in  salt. 


V]  OF   THE   SUN-ANIMALCULES  265 

of  spheres.  Outwards  through  the  protoplasm,  and  stretching  far 
beyond  the  spherical  surface  of  the  cell,  there  run  stiff  linear 
threads  of  modified  or  differentiated  protoplasm,  replaced  or 
reinforced  in  some  cases  by  delicate  sihceous  needles.  In  either 
case  we  know  little  or  nothing  about  the  forces  which  lead  to  their 
production,  and  we  do  not  hide  our  ignorance  when  we  ascribe 
their  development  to  a  "radial  polarisation"  of  the  cell.  In  the 
case  of  the  protoplasmic  filament,  we  may  (if  we  seek  for  a 
hypothesis),  suppose  that  it  is  somehow  comparable  to  a  viscid 
stream,  or  "liquid  vein,"  thrust  or  squirted  out  from  the  body  of 
the  cell.  But  when  it  is  once  formed,  this  long  and  comparatively 
rigid  filament  is  separated  by  a  distinct  surface  from  the  neigh- 
bouring protoplasm,  that  is  to  say  from  the  more  fluid  surface- 
protoplasm  of  the  cell ;  and  the  latter  begins  to  creep  up  the 
filament,  just  as  water  would  creep  up  the  interior  of  a  glass  tube, 
or  the  sides  of  a  glass  rod  immersed  in  the  liquid.  It  is  the  simple 
case  of  a  balance  between  three  separate  tensions  :  ( 1 )  that  between 
the  filament  and  the  adjacent  protoplasm,  (2)  that  between  the 
filament  and  the  adjacent  water,  and  (3)  that  between  the  water 
and  the  protoplasm.  Calling  these  tensions  respectively  Tf^.,,  Tf^, 
and  T^,p,  equilibrium  will  be  attained  when  the  angle  of  contact 
between    the    fluid   protoplasm   and   the   filament   is   such   that 

T     —  T 

cos  a  =     ^^-, -^ .     It  is  evident  in  this  case  that  the  angle  is 


T 


fp 


a  very  small  one.  The  precise  form  of  the  curve  is  somewhat 
different  from  that  which,  under  ordinary  circumstances,  is  assumed 
by  a  liquid  which  creeps  up  a  solid  surface,  as  water  in  contact 
with  air  creeps  up  a  surface  of  glass ;  the  difference  being  due  to 
the  fact  that  here,  owing  to  the  density  of  the  protoplasm  being 
practically  identical  with  that  of  the  surrounding  medium,  the 
whole  system  is  practically  immune  from  gravity.  Under  normal 
circumstances  the  curve  is  part  of  the  "elastic  curve"  by  which 
that  surface  of  revolution  is  generated  which  we  have  called, 
after  Plateau,  the  nodoid ;  but  in  the  present  case  it  is  apparently 
a  catenary.  Whatever  curve  it  be,  it  obviously  forms  a  surface 
of  revolution  around  the  filament. 

Since    the    attraction    exercised    by    this    surface    tension    is 
symmetrical  around  the  filament,  the  latter  will  be  pulled  equally 


266 


THE   FORMS   OF  CELLS 


[CH. 


in  all  directions ;  in  other  words  it  will  tend  to  be  set  normally 
to  the  surface  of  the  sphere,  that  is  to  say  radiating  directly 
outwards  from  the  centre.  If  the  distance  between  two  adjacent 
filaments  be  considerable,  the  curve  will  simply  meet  the  filament 
at  the  angle  a  already  referred  to ;  but  if  they  be  sufficiently  near 
together,  we  shall  have  a  continuous  catenary  curve  forming  a 
hanging  loop  between  one  filament  and  the  other.  And  when  this 
is  so,  and  the  radial  filaments  are  more  or  less  symmetrically 
interspaced,  we  may  have  a  beautiful  system  of  honeycomb-like 
depressions  over  the  surface  of  the  organism,  each  cell  of  the 
honeycomb  having  a  strictly  defined  geometric  configuration. 

In  the  simpler  Eadiolaria,  the  spherical  form  of  the  entire 
organism  is  equally  well-marked ;  and  here,  as  also  in  the  more 
complicated  Heliozoa  (such  as  Actinosphaerium),  the  organism  is 


Fig.  90.     A,  Trypanosoma  tineae  (after  Minchin);    B,  Spirochaeta  anodontae 
(after  Fantham). 

differentiated  into  several  distinct  layers,  each  boundary  surface 
tending  to  be  spherical,  and  so  constituting  sphere  within  sphere. 
One  of  these  layers  at  least  is  close  packed  with  vacuoles,  forming 
an  "alveolar  mesh  work,"  with  the  configurations  of  which  we  shall 
attempt  in  another  chapter  to  correlate  the  characteristic  structure 
of  certain  complex  types  of  skeleton. 


An  exceptional  form  of  cell,  but  a  beautiful  manifestation  of 
surface-tension  (or  so  I  take  it  to  be),  occurs  in  Trypanosomes,  those 
tiny  parasites  of  the  blood  that  are  associated  with  sleeping- 
sickness  and  many  other  grave  or  dire  maladies.  These  tiny 
organisms  consist  of  elongated  solitary  cells  down  one  side  of  which 
runs  a  very  delicate  frill,  or  "undulating  membrane,"  the  free 
edge  of  which  is  seen  to  be  slightly  thickened,  and  the  whole  of 


V] 


OF  UNDULATINC4  MEMBRANES 


267 


which  undergoes  rhythmical  and  beautiful  wavy  movements. 
When  certain  Trypanosomes  are  artificially  cultivated  (for  instance 
T.  rotatorium,  from  the  blood  of  the  frog),  phases  of  growth  are 
witnessed  in  which  the  organism  has  no  undulating  membrane, 
but  possesses  a  long  cilium  or  "flagellum,'"  springing  from  near 
the  front  end,  and  exceeding  the  whole  body  in  length*.  Again, 
in  T.  lewisii,  when  it  reproduces  by  "multiple  fission,"  the 
products  of  this  division  are  likewise  devoid  of  an  undulating 
membrane,  but  are  provided  with  a  long  free  flagellum  "j".     It  is 


Fig.  91.     A,  Trichomonas  muris,  Hartmann;    B,  Trichomastix  serpentis,  Dobell; 
C,  Trichomonas  angusta,  Alexeieff.     (After  Kofoid.) 

a  plausible  assumption  to  suppose  that,  as  the  flagellum  waves 
about,  it  comes  to  lie  near  and  parallel  to  the  body  of  the  cell. 
and  that  the  frill  or  undulating  membrane  is  formed  by  the  clear, 
fluid  protoplasm  of  the  surface  layer  springing  up  in  a  film  to  run 
up  and  along  the  flagellum,  just  as  a  soap-film  would  be  formed  in 
similar  circumstances. 

This  mode  of  formation  of  the  undulating  membrane  or  frill 
appears  to  be  confirmed  by  the  appearances  shewn  in  Fig.   91. 

*  Cf.  Doflein,  Lehrbuch  der  Protozoenkmide,  1911,  p.  422. 

t  Cf.  Minchin,  Introduction  to  the  Study  of  the  Protozoa,  1914    p.  293,  Fig.  127. 


268 


THE   FORMS   OF   CELLS 


[CH. 


Here  we  have  three  little  organisms  closely  allied  to  the  ordinary 
Trypanosomes,  of  which  one,  Trichomastix  (B),  possesses  four 
flagella,  and  the  other  two.  Trichomonas,  apparently  three  only: 
the  two  latter  possess  the  frill,  which  is  lacking  in  the  first*.  But 
it  is  impossible  to  doubt  that  when  the  frill  is  present  (as  in  A  and 
C),  its  outer  edge  is  constituted  by  the  apparently  missing  flagellum 
{a),  which  has  become  attached  to  the  body  of  the  creature  at  the 
point  c,  near  its  posterior  end ;  and  all  along  its  course,  the  super- 
ficial protoplasm  has  been  drawn  out  into  a  film,  between  the 
flagellum  {a)  and  the  adjacent  surface  or  edge  of  the  body  (6). 

Moreover,  this  mode  of  formation  has  been  actually  witnessed 
and  described,  though  in  a  somewhat  exceptional  case.  The  little 
flagellate  monad  Herpetomonas  is  normally  destitute  of  an  undulat- 
ing membrane,  but  possesses  a  single  long  terminal  flagellum. 
According  to  Dr  D.  L.  Mackinnon,  the  cytoplasm  in  a  certain  stage 
of  growth  becomes  somewhat  "sticky,"  a  phrase  which  we  may 
in  all  probability  interpret  to  mean  that  its  surface  tension  is 

being  reduced.  For  this  stickiness  is 
shewn  in  two  ways.  In  the  first  place, 
the  long  body,  in  the  course  of  its 
various  bending  movements,  is  apt  to 
adhere  head  to  tail  (so  to  speak),  giving 
a  rounded  or  sometimes  annular  form 
to  the  organism,  such  as  has  also  been 
described  in  certain  species  or  stages 
of  Trypanosomes.  But  again,  the 
long  flagellum,  if  it  get  bent  back- 
wards upon  the  body,  tends  to  adhere 
to  its  surface.  "Where  the  flagellum 
was  pretty  long  and  active,  its  efforts 
to  continue  movement  under  these 
abnormal  conditions  resulted  in  the 
gradual  lifting  up  from  the  cytoplasm 
of  the  body  of  a  sort  of  fseudo- 
undulating  membrane  (Fig.  92).  The  movements  of  this  structure 
were  so  exactly  those  of  a  true  undulating  membrane  that  it  was 

*  Cf.  C.  A.  Kofoid  and  Olive  Swezy,  On  Trichomonad  Flagellates,  etc.  Pr. 
Amer.  Acad,  of  Arts  and  Sci.  li,  pp.  289-378,  1915. 


Fig.  92.  Herpetomonas  assuming 
the  undulatory  membrane  of'  a 
Trypanosome.  (After  D.  L. 
Mackinnon. ) 


V]  OF   UNDULATING   MEMBRANES  269 

difficult  to  believe  one  was  not  dealing  with  a  small,  blunt 
trypanosome*.''  This  in  short  is  a  precise  description  of  the 
mode  of  development  which,  from  theoretical  considerations 
alone,  we  should  conceive  to  be  the  natural  if  not  the  only 
possible  way  in  which  the  undulating  membrane  could  come  into 
existence. 

There  is  a  genus  closely  allied  to  Trypanosoma,  viz.  Trypano- 
plasma,  which  possesses  one  free  flagellum,  together  with  an 
undulating  membrane ;  and  it  resembles  the  neighbouring  genus 
Bodo,  save  that  the  latter  has  two  fiagella  and  no  undulating 
membrane.  In  like  manner,  Trypanosoma  so  closely  resembles 
Herpetomonas  that,  when  individuals  ascribed  to  the  former  genus 
exhibit  a  free  flagellum  only,  they  are  said  to  be  in  the  "Her- 
petomonas stage."  In  short  all  through  the  order,  we  have  pairs 
of  genera,  which  are  presumed  to  be  separate  and  distinct,  viz. 
Trypanosoma-Herpetomonas,  Trypanoplasma-Bodo,Trichomastix- 
Trichomonas,  in  which  one  differs  from  the  other  mainly  if  not 
solely  in  the  fact  that  a  free  flagellum  in  the  one  is  replaced  by  an 
undulating  membrane  in  the  other.  We  can  scarcely  doubt  that 
the  two  structures  are  essentially  one  and  the  same. 

•  The  undulating  membrane  of  a  Trypanosome,  then,  according 
to  our  interpretation  of  it,  is  a  liquid  film  and  must  obey  the  law 
of  constant  mean  curvature.  It  is  under  curious  limitations  of 
freedom :  for  by  one  border  it  is  attached  to  the  comparatively 
motionless  body,  while  its  free  border  is  constituted  by  a  flagellum 
which  retains  its  activity  apd  is  being  constantly  thrown,  like  the 
lash  of  a  whip,  into  wavy  curves.  It  follows  that  the  membrane, 
for  every  alteration  of  its  longitudinal  curvature,  must  at  the  same 
instant  become  curved  in  a  direction  perpendicular  thereto ;  it 
bends,  not  as  a  tape  bends,  but  with  the  accompaniment  of  beautiful 
but  tiny  waves  of  double  curvature,  all  tending  towards  the 
establishment  of  an  "  equipotential  surface  "  ;  and  its  characteristic 
undulations  are  not  originated  by  an  active  mobility  of  the 
membrane  but  are  due  to  the  molecular  tensions  which  produce 
the  very  same  result  in  a  soap-film  under  similar  circumstances. 
In  certain  Spirochaetes,  S.  anodontae  (Fig.  90)  and  S.  halhiani 

*  D.   L.   Mackinnon,   Herpetomonads  from  the  Alimentary  Tract  of   certain 
Dungtlies,  Parasitoloyy,  ni,  p.  268,  1910. 


270  THE   FORMS   OF   CELLS  [ch. 

(which  we  find  in  oysters),  a  very  similar  undulating  membrane 
exists,  but  it  is  coiled  in  a  regular  spiral  round  the  body  of  the  cell. 
It  forms  a  "screw-surface,"  or  helicoid,  and,  though  we  might 
think  that  nothing  could  well  be  more  curved,  yet  its  mathematical 
properties  are  such  that  it  constitutes  a  "ruled  surface"  whose 
"mean  curvature"  is  everywhere  nil;  and  this  property  (as  we 
have  seen)  it  shares  with  the  plane,  and  with  the  plane  alone. 
Precisely  such  a  surface,  and  of  exquisite  beauty,  may  be 
produced  by  bending  a  wire  upon  itself  so  that  part  forms  an 
axial  rod  and  part  a  spiral  wrapping  round  the  axis,  and  then 
dipping  the  whole  into  a  soapy  solution. 

These  undulating  and  helicoid  surfaces  are  exactly  reproduced 
among  certain  forms  of  spermatozoa.  The  tail  of  a  spermatozoon 
consists  normally  of  an  axis  surrounded  by  clearer  and  more  fluid 
protoplasm,  and  the  axis  sometimes  splits  up  into  two  or  more 
slender  filaments.  To  surface  tension  operating  between  these 
and  the  surface  of  the  fluid  protoplasm  (just  as  in  the  case  of  the 
fiagellum  of  the  Trypanosome),  I  ascribe  the  formation  of  the 
undulating  membrane  which  we  find,  for  instance,  in  the  spermato- 
zoa of  the  newt  or  salamander;  and  of  the  helicoid  membrane, 
wrapped  in  a  far  closer  and  more  beautiful  spiral  than  that  which 
we  saw  in  Spirochaeta,  which  is  characteristic  of  the  spermatozoa 
of  many  birds. 

Before  we  pass  from  the  subject  of  the  conformation  of  the 
solitary  cell  we  must  take  some  account  of  certain  other  exceptional 
forms,  less  easy  of  explanation,  and  still  less  perfectly  understood. 
Such  is  the  case,  for  instance,  with  the  red  blood-corpuscles  of  man 
and  other  vertebrates ;  and  among  the  sperm-cells  of  the  decapod 
Crustacea  we  find  forms  still  more  aberrant  and  not  less  perplexing. 
These  are  among  the  comparatively  few  cells  or  cell-like  structures 
whose  form  seenis  to  be  incapable  of  explanation  by  theories  of 
surface-tension. 

In  all  the  mammalia  (save  a  very  few)  the  red  blood-corpuscles 
are  flattened  circular  discs,  dimpled  in  upon  their  two  opposite 
sides.  This  configuration  closely  resembles  that  of  an  india- 
rubber  ball  when  we  pinch  it  tightly  between  finger  and  thumb ; 
and  we  may  also  compare  it  with  that  experiment  of  Plateau's 


V]  OF   BLOOD   CORPUSCLES  271 

(described  on  p.  223),  where  a  flat  cylindrical  oil-drop,  of  certain 
relative  dimensions,  can,  by  sucking  away  a  little  of  the  contained 
oil,  be  made  to  assume  the  form  of  a  biconcave  disc,  whose  periphery 
is  part  of  a  nodoidal  surface.  From  the  relation  of  the  nodoid 
to  the  "elastic  curve,"  we  perceive  that  these  two  examples  are 
closely  akin  one  to  the  other. 

The  form  of  the  corpuscle  is  symmetrical,  and  its  surface  is 
a  surface  of  revolution ;  but  it 
is  obviously  not  a  surface  of 
constant  mean  curvature,  nor  of 
constant  pressure.  For  we  see 
at  once  that,  in  the  sectional 
diagram  (Fig.  93),  the  pressure 
inwards  due  to  surface  tension 
is  positive  at  A,  and  negative  at  C ;  at  B  there  is  no 
curvature  in  the  plane  of  the  paper,  while  perpendicular  to 
it  the  curvature  is  negative,  and  the  pressure  therefore  is  also 
negative.  Accordingly,  from  the  point  of  view  of  surface  tension 
alone,  the  blood-corpuscle  is  not  a  surface  of  equilibrium ;  or  in 
other  words,  it  is  not  a  fluid  drop  suspended  in  another  liquid. 
It  is  obvious  therefore  that  some  other  force  or  forces  must  be 
at  work,  and  the  simple  effect  of  mechanical  pressure  is  here 
excluded,  because  the  blood-corpuscle  exhibits  its  characteristic 
shape  while  floating  freely  in  the  blood.  In  the  lower  vertebrates 
the  blood-corpuscles  have  the  form  of  a  flattened  oval  disc,  with 
rather  sharp  edges  and  ellipsoidal  surfaces,  and  this  again  is 
manifestly  not  a  surface  of  equilibrium. 

Two  facts  are  especially  noteworthy  in  connection  with  the 
form  of  the  blood-corpuscle.  In  the  first  place,  its  form  is  only 
maintained,  that  is  to  say  it  ig  only  in  equilibrium,  in  relation  to 
certain  properties  of  the  medium  in  which  it  floats.  If  we  add  a 
little  water  to  the  blood,  the  corpuscle  quickly  loses  its  character- 
istic shape  and  becomes  a  spherical  drop,  that  is  to  say  a  true 
surface  of  minimal  area  and  of  stable  equilibrium.  If  on  the  other 
hand  we  add  a  strong  solution  of  salt,  or  a  little  glycerine,  the 
corpuscle  contracts,  and  its  surface  becomes  puckered  and  uneven. 
In  these  phenomena  it  is  so  far  obeying  the  laws  of  diffusion  and 
of  surface  tension. 


272  THE   FORMS   OF   CELLS  [ch. 

In  the  second  place,  it  can  be  exactly  imitated  artificially  by 
means  of  other  colloid  substances.  Many  years  ago  Norris  made  the 
very  interesting  observation  that  in  an  emulsion  of  glue  the  drops 
assumed  a  biconcave  form  resembling  that  of  the  mammalian  cor- 
puscles*. The  glue  was  impure,  and  doubtless  contained  lecithin  ; 
and  it  is  possible  (as  Professor  Waymouth  Reid  tells  me)  to  make 
a  similar  emulsion  with  cerebrosides  and  cholesterin  oleate,  in 
which  the  same  conformation  of  the  drops  or  particles  is  beautifully 
shewn.  Now  such  cholesterin  bodies  have  an  important  place 
among  those  in  which  Lehmann  and  others  have  shewn  and  studied 
the  formation  of  fluid  crystals,  that  is  to  say  of  bodies  in  which 
the  forces  of  crystallisation  and  the  forces  of  surface  tension  are 
battling  with  one  another  f ;  and,  for  want  of  a  better  explanation, 
we  may  in  the  meanwhile  suggest  that  some  such  cause  is  at  the 
bottom  of  the  conformation  the  explanation  of  which  presents  so 
many  difficulties.  But  we  must  not,  perhaps,  pass  from  this 
subject  without  adding  that  the  case  is  a  difficult  and  complex 
one  from  the  physiological  point  of  view.  For  the  surface  of  a 
blood-corpuscle  consists  of  a  "semi-permeable  membrane,"'  through 
which  certain  substances  pass  freely  and  not  others  (for  the  most 
part  anions  and  not  cations),  and  it  may  be,  accordingly,  that  we 
have  in  life  a  continual  state  of  osmotic  inequilibrium,  of  negative 
osmotic  tension  within,  to  which  comparatively  simple  cause  the 
imperfect  distension  of  the  corpuscle  may  be  also  due  J.  The  whole 
phenomenon  would  be  comparatively  easy  to  understand  if  we 
might  postulate  a  stifEer  peripheral  region  to  the  corpuscle,  in  the 
form  for  instance  of  a  peripheral  elastic  ring.  Such  an  annular 
thickening  or  stiffening,  like  the  "  collapse-rings ''  which  an  engineer 
inserts  in  a  boiler,  has  been  actually  asserted  to  exist,  but  its 
presence  is  not  authenticated. 

But  it  is  not  at  all  improbable  that  we  have  still  much  to 
learn  about  the  phenomena  of  osmosis  itself,  as  manifested  in  the 
case  of  m^ute  bodies  such  as  a  blood-corpuscle ;  and  (as  Professor 
Peddie  suggests  to  me)  it  is  by  no  means  impossible  that  curvature 

*  Prnc  B  y.  Soc.  xii,  pp.  251-257,  1862-3. 

t  Cf.  (int.  al.)  Lehmann,  Ueber  scheinbar  lebende  KristaUe  und  Myelinformen, 
Arch.  f.  Enho.  Mech.  xxvi,  p.  483,  1908;  Ann.  d.  Physik,  xliv,  p   969,  1914. 

f  Cf.  B.  Moore  and  H.  C.  Roaf,  On  the  Osmotic  Equilibrium  of  the  Red 
Blood  Corpuscle,  BiocJiem.  Journal,  in,  p.  55,  1908. 


V] 


OF  CEETAIN   OSMOTIC  PHENOMENA 


273 


of  the  surface  may  itself  modify  the  osmotic  or  perhaps  the  adsorp- 
tive  action.  If  it  should  be  found  that  osmotic  action  tended  to 
stop,  or  to  reverse,  on  change  of  curvature,  it  would  follow  that 
this  phenomenon  would  give  rise  to  internal  currents ;  and  the 
change  of  pressure  consequent  on  these  would  tend  to  intensify 
the  change  of  curvature  when  once  started*. 

The  sperm-cells  of  the  Decapod  Crustacea  exhibit  various 
singular  shapes.  In  the  Crayfish  they  are  flattened  cells  with 
stiff  curved  processes  radiating  outwards  like  a  St  Catherine's 
wheel ;  in  Inachus  there  are  two  such  circles  of  stif?  processes ; 
in  Galathea  we  have  a  still  more  complex  form,  with  long  and 


Fig.  94.     Sperm-cells  of  Decapod  Crustacea  (after  Koltzoff).     a,  Inachus  scorpio; 
b,  Galathea  squamifera ;    c,  do.  after  maceration,  to  shew  spiral  fibrillae. 

slightly  twisted  processes.  In  all  these  cases,  just  as  in  the  case 
of  the  blood-corpuscle,  the  structure  alters,  and  finally  loses,  its 
characteristic  form  when  the  nature  or  constitution  (or  as  we  may 
assum.e  in  particular — the  density)  of  the  surrounding  medium  is 
changed. 

Here  again,  as  in  the  blood-corpuscle,  we  have  to  do  with  a 
very  important  force,  which  we  had  not  hitherto  considered  in  this 
connection, — the  force  of  osmosis,  manifested  under  conditions 
similar  to  those  of  Pfeffer's  classical  experiments  on  the  plant-cell. 
The  surface  of  the  cell  acts  as  a  "semi-permeable  membrane," 

*  For  an  attempt  to  explain  the  form  of  a  blood-corpuscle  by  surface-tension 
alone,  see  Rice,  Phil.  Mag.  Nov.  1914;   but  cf.  Shorter,  ibid.  Jan.  1915. 

T.  G.  18 


274 


THE   FORMS   OF   CELLS 


[CH. 


permitting  the  passage  of  certain  dissolved  substances  (or  their 
"ions")  and  including  or  excluding  others;  and  thus  rendering 
manifest  and  measurable  the  existence  of  a  definite  "osmotic 
pressure."  Li  the  case  of  the  sperm-cells  of  Inachus,  certain 
quantitative  experiments  have  been  performed*.  The  sperm-cell 
exhibits  its  characteristic  conformation  while  lying  in  the  serous 
fluid  of  the  animal's  body,  in  ordinary  sea-water,  or  in  a  5  per 
cent,  solution  of  potassium  nitrate ;  these  three  fluids  being  all 
"isotonic"  with  one  another.  As  we  alter  the  concentration  of 
potassium  nitrate,  the  cell  assumes  certain  definite  forms  corre- 
sponding to  definite  concentrations  of  the  salt;  and,  as  a  further 
and  final  proof  that  the  phenomenon  is  entirely  physical,  it  is 
found  that  other  salts  produce  an  identical  effect  when  their 
concentration  is  proportionate  to   their  molecular  weight,   and 


KNO 


Pig.  95.     Sperm-cells  of  Inachus,  as  they  appear  in  saline  solutions  of 
varying  density.     (After  Koltzoff.) 

whatever  identical  effect  is  produced  by  various  salts  in  their 
respective  concentrations,  a  similarly  identical  effect  is  produced 
when  these  concentrations  are  doubled  or  otherwise  proportionately 
changed  f. 

Thus  the  following  table  shews  the  percentage  concentrations 
of  certain  salts  necessary  to  bring  the  cell  into  the  forms  a  and  c 
of  Fig.  95 ;  in  each  case  the  quantities  are  proportional  to  the 
molecular  weights,  and  in  each  case  twice  the  quantity  is  necessary 
to  produce  the  effect  of  Fig.  95  c  compared  with  that  which  gives 
rise  to  the  all  but  spherical  form  of  Fig.  95  a. 

*  Koltzoff,  N.  K. ,  Studien  iiber  die  Gestalt  der  Zelle,  Arch.  f.  mihrosk.  Anaf. 
Lxvn,  pp.  364-571,  1905;  Biol.  Centralbl.  xxm,  pp.  680-696,  1903,  xxvi, 
pp.  854-863,  1906;  Arch.  f.  Zellforschung,  ii,  pp.  1-65,  1908,  vii,  pp.  344-423, 
1911;    Anat.  Anzeiger,  xli,  pp.  183-206,  1912. 

t  Cf.  supra,  p.  129. 


V]  OF  CERTAIN  OSMOTIC   PHENOMENA  275 

%  concentration  of  salts  in  wkicli 

the  sperm-cell  of  Inachus 

assumes  the  form  of 


%.  a 

fig.  c 

Sodium  chloride 

0-6 

1-2 

Sodium  nitrate 

0-85 

1-7 

Potassium  nitrate 

1-0 

2-0 

Acetic  acid 

2-2 

4-5 

Cane  sugar 

5-0 

100 

If  we  look  then,  upon  the  spherical  form  of  the  cell  as  its  true 
condition  of  symmetry  and  of  equilibrium,  we  see  that  what  we 
call  its  normal  appearance  is  just  one  of  many  intermediate  phases 
of  shrinkage,  brought  about  by  the  abstraction  of  fluid  from  its 
interior  as  the  result  of  an  osmotic  pressure  greater  outside  than 
inside  the  cell,  and  where  the  shrinkage  of  volume  is  not  kept 
pace  with  by  a  contraction  of  the  surface-area.  In  the  case  of  the 
blood-corpuscle,  the  shrinkage  is  of  no  great  amount,  and  the 
resulting  deformation  is  symmetrical ;  such  structural  inequality 
as  may  be  necessary  to  account  for  it  need  be  but  small.  But 
in  the  case  of  the  sperm-cells,  we  must  have,  and  we  actually  do 
find,  a  somewhat  complicated  arrangement  of  more  or  less  rigid 
or  elastic  structures  in  the  wall  of  the  cell,  which  like  the  wire 
framework  in  Plateau's  experiments,  restrain  and  modify  the 
forces  acting  on  the  drop.  In  one  form  of  Plateau's  experiments, 
instead  of  supporting  his  drop  on  rings  or  frames  of  wire,  he  laid 
upon  its  surface  one  or  more  elastic  coils;  and  then,  on  with- 
drawing oil  from  the  centre  of  his  globule,  he  saw  its  uniform 
shrinkage  counteracted  by  the  spiral  springs,  with  the  result  that 
the  centre  of  each  elastic  coil  seemed  to  shoot  out  into  a  prominence. 
Just  such  spiral  coils  are  figured 
(after  KoltzofE)  in  Fig.  96 ;  and  they 
may  be  regarded  as  precisely  akin  to 
those  local  thickenings,  spiral  and 
other,  to  which  we  have  already 
ascribed  the  cylindrical  form  of  the 
Spirogyra  cell.  In  all  probabihty  we 
must  in  like  manner  attribute  the 

peculiar  spiral  and  other  forms,  for      -p-    nc     o    '        n   j;  r> 

^  ^  '  i<  ig.  96.     bperm-cell  of  Dromia. 

instance  of  many  Infusoria,  to  the  (After  KoltzofE.) 

18—2 


276  THE   FORMS   OF   CELLS  [ch.  v 

presence,  among  the  multitudinous  other  dift'erentiations  of  their 
protoplasmic  substance,  of  such  more  or  less  elastic  fibrillae, 
which  play  as  it  were  the  part  of  a  microscopic  skeleton*. 

But  these  cases  which  we  have  just  dealt  with,  lead  us  to 
another  consideration.  In  a  semi-permeable  membrane,  through 
which  water  passes  freely  in  and  out,  the  conditions  of  a  liquid 
surface  are  greatly  modified ;  and,  in  the  ideal  or  ultimate  case, 
there  is  neither  surface  nor  surface  tension  at  all.  And  this  would 
lead  us  somewhat  to  reconsider  our  position,  and  to  enquire 
whether  the  true  surface  tension  of  a  liquid  film  is  actually 
responsible  for  all  that  we  have  ascribed  to  it,  or  whether  certain 
of  the  phenomena  which  we  have  assigned  to  that  cause  may  not 
in  part  be  due  to  the  contractility  of  definite  and  elastic  membranes. 
But  to  investigate  this  question,  in  particular  cases,  is  rather  for 
the  physiologist :  and  the  morphologist  may  go  on  his  way, 
paying  little  heed  to  what  is  no  doubt  a  difficulty.  In  surface 
tension  we  have  the  production  of  a  film  with  the  properties  of  an 
elastic  membrane,  and  with  the  special  peculiarity  that  contraction 
continues  with  the  same  energy  however  far  the  process  may  have 
already  gor^e ;  while  the  ordinary  elastic  membrane  contracts  to 
a  certain  extent,  and  contracts  no  more.  But  within  wide  limits 
the  essential  phenomena  are  the  same  in  both  cases.  Our 
fundamental  equations  apply  to  both  cases  alike.  And  accord- 
ingly, so  long  as  our  purpose  is  mofphological,  so  long  as  what  we 
seek  to  explain  is  regularity  and  definiteness  of  form,  it  matters 
little  if  we  should  happen,  here  or  there,  to  confuse  surface  tension 
with  elasticity,  the  contractile  forces  manifested  at  a  liquid 
surface  with  those  which  come  into  play  at  the  complex  internal 
surfaces  of  an  elastic  solid. 

*  As  Bethe  points  out  (Zellgestalt,  Plateausche  Fliissigkeitsfigur  unci  Neuro- 
fibrille,  Anat.  Anz.  XL.  p.  209,  1911),  the  spiral  fibres  of  which  Koltzoff  speaks  must 
lie  in  the  surface,  and  not  within  the  substance,  of  the  cell  whose  conformation  is 
affected  by  them. 


CHAPTER    VI 

A  NOTE  OX  ADSORPTION 

A  very  important  corollary  to,  or  amplification  of  the  theory 
of  surface  tension  is  to  be  found  in  the  modern  chemico-physical 
doctrine  of  Adsorption*.  In  its  full  statement  this  subject  soon 
becomes  complicated,  and  involves  physical  conceptions  and 
mathematical  treatment  which  go  beyond  our  range.  But  it  is 
necessary  for  us  to  take  account  of  the  phenomenon,  though  it 
be  in  the  most  elementary  way. 

In  the  brief  account  of  the  theory  of  surface  tension  with  which 
our  last  chapter  began,  it  was  pointed  out  that,  in  a  drop  of  liquid, 
the  potential  energy  of  the  system  could  be  diminished,  and  work 
manifested  accordingly,  in  two  ways.  In  the  first  place  we  saw 
that,  at  our  liquid  surface,  surface  tension  tends  to  set  up  an 
equilibrium  of  form,  in  which  the  surface  is  reduced  or  contracted 
either  to  the  absolute  minimum  of  a  sphere,  or  at  any  rate  to  the 
least  possible  area  which  is  permitted  by  the  various  circumstances 
and  conditions ;  and  if  the  two  bodies  which  comprise  our  system, 
namely  the  drop  of  liquid  and  its  surrounding  medium,  be  simple 
substances,  and  the  system  be  uncomplicated  by  other  distributions 
of  force,  then  the  energy  of  the  system  will  have  done  its  work 
when  this  equilibrium  of  form,  this  minimal  area  of  surface,  is 
once  attained.  This  phenomenon  of  the  production  of  a  minimal 
surface-area  we  have  now  seen  to  be  of  fundamental  importance 
in  the  external  morphology  of  the  cell,  and  especially  (so  far 
as  we  have  yet  gone)  of  the  solitary  cell  or  unicellular  organism. 

*  See  for  a  further  but  still  elementary  account,  Michaelis,  Dynamics  of  Surfaces, 
1914,  p.  22  seq.;  Macallum,  Oberfldchenspcmnttng  und  Lebenserscheinungen,  in 
Asher-Spiro's  Ergebnisse  der  Physiologic,  xi,  pp.  598-658,  1911;  see  also  W.  W. 
Taylor's  Chemistry  of  Colloids,  1915,  p.  221  seq.,  Wolfgang  Ostwald,  Grundriss  der 
Kolloidchemie,  1909,  and  other  text-books  of  physical  chemistry;  and  Bayhss's 
Principles  of  General  Physiology,  pp.  54-73,  1915. 


278  A  NOTE   ON   ADSORPTION  [ch. 

But  we  also  saw,  according  to  Gauss's  equation,  that  the 
potential  energy  of  the  system  will  be  diminished  (and  its  diminu- 
tion will  accordingly  be  manifested  in  work)  if  from  any  cause 
the  specific  surface  energy  be  diminished,  that  is  to  say  if  it  be 
brought  more  nearly  to  an  equality  with  the  specific  energy  of  the 
molecules  in  the  interior  of  the  liquid  mass.  This  latter  is  a 
phenomenon  of  great  moment  in  modern  physiology,  and,  while 
we  need  not  attempt  to  deal  with  it  in  detail,  it  has  a  bearing  on 
cell-form  and  cell-structure  which  we  cannot  afford  to  overlook. 

In  various  ways  a  diminution  of  the  surface  energy  may  be 
brought  about.  For  instance,  it  is  known  that  every  isolated  drop 
of  fluid  has,  under  normal  circumstances,  a  surface-charge  of 
electricity :  in  such  a  way  that  a  positive  or  negative  charge  (as 
the  case  may  be)  is  inherent  in  the  surface  of  the  drop,  while  a 
corresponding  charge,  of  contrary  sign,  is  inherent  in  the 
immediately  adjacent  molecular  layer  of  the  surrounding  medium. 
Now  the  effect  of  this  distribution,  by  which  all  the  surface 
molecules  of  our  drop  are  similarly  charged,  is  that  by  virtue  of 
this  charge  they  tend  to  repel  one  another,  and  possibly  also  to 
draw  other  molecules,  of  opposite  charge,  from  the  interior  of  the 
mass ;  the  result  being  in  either  case  to  antagonise  or  cancel, 
more  or  less,  that  normal  tendency  of  the  surface  molecules  to 
attract  one  another  which  is  manifested  in  surface  tension.  In 
other  words,  an  increased  electrical  charge  concentrating  at  the 
surface  of  a  drop  tends,  whether  it  be  positive  or  negative,  to 
lower  the  surface  tension. 

But  a  still  more  important  case  has  next  to  be  considered. 
Let  us  suppose  that  our  drop  consists  no  longer  of  a  single  chemical 
substance,  but  contains  other  substances  either  in  suspension  or 
in  solution.  Suppose  (as  a  very  simple  case)  that  it  be  a  watery 
fluid,  exposed  to  air,  and  containing  droplets  of  oil :  we  know  that 
the  specific  surface  tension  of  oil  in  contact  with  air  is  much  less 
than  that  of  water,  and  it  follows  that,  if  the  watery  surface  of 
our  drop  be  replaced  by  an  oily  surface  the  specific  surface  energy 
of  the  system  will  be  notably  diminished.  Now  under  these 
circumstances  it  is  found  that  (quite  apart  from  gravity,  by  which 
the  oil  might  float  to  the  surface)  the  oil  has  a  tendency  to  be 
drawn  to  the  surface ;  and  this  phenomenon  of  molecular  attraction 


VI]  ON   SURFACE-CONCENTRATION  279 

or  "adsorption"  represents  the  work  done,  equivalent  to  the 
diminished  potential  energy  of  the  system*.  In  more  general 
terms,  if  a  liquid  (or  one  or  other  of  two  adjacent  liquids)  be  a 
chemical  mixture,  some  one  constituent  in  which,  if  it  entered 
into  or  increased  in  amount  in  the  surface  layer,  would  have  the 
effect  of  diminishing  its  surface  tension,  then  that  constituent  will 
have  a  tendency  to  accumulate  or  concentrate  at  the  surface :  the 
surface  tension  may  be  said,  as  it  were,  to  exercise  an  attraction 
on  this  constituent  substance,  drawing  it  into  the  surface  layer, 
and  this  tendency  will  proceed  until  at  a  certain  "surface  con- 
centration" equilibrium  is  reached,  its  opponent  being  that  osmotic 
force  which  tends  to  keep  the  substance  in  uniform  solution  or 
diffusion. 

In  the  complex  mixtures  which  constitute  the  protoplasm  of 
the  living  cell,  this  phenomenon  of  "adsorption"  has  abundant 
play  :  for  many  of  these  constituents,  such  as  oils,  soaps,  albumens, 
etc.  possess  the  required  property  of  diminishing  surface  tension. 

Moreover,  the  more  a  substance  has  the  power  of  lowering  the 
surface  tension  of  the  Uquid  in  which  it  happens  to  be  dissolved, 
the  more  will  it  tend  to  displace  another  and  less  effective  substance 
from  the  surface  layer.  Thus  we  know  that  protoplasm,  always 
contains  fats  or  oils,  not  only  in  visible  drops,  but  also  in  the 
finest  suspension  or  "colloidal  solution."  If  under  any  impulse, 
such  for  instance  as  might  arise  from  the  Brownian  movement, 
a  droplet  of  oil  be  brought  close  to  the  surface,  it  is  at  once  drawn 
into  that  surface,  and  tends  to  spread  itself  in  a  thin  layer  over 
the  whole  surface  of  the  cell.  But  a  soapy  surface  (for  instance) 
would  have  in  contact  with  the  surrounding  water  a  surface  tension 
even  less  than  that  of  the  film  of  oil :  and  consequently,  if  soap 
be  present  in  the  water  it  will  in  turn  be  adsorbed,  and  will  tend 
to  displace  the  oil  from  the  surface  pellicle  f.     And  this  is  all  as 

*  The  first  instance  of  what  we  now  call  an  adsorptive  phenomenon  was 
observed  in  soap-bubbles.  Leidenfrost,  in  1756,  was  aware  that  the  outer  layer 
of  the  bubble  was  covered  by  an  "oily"  layer.  A  hundred  years  later  Dupre 
shewed  that  in  a  soap-solution  the  soap  tends  to  concentrate  at  the  surface,  so 
that  the  surface-tension  of  a  very  weak  solution  is  very  httle  different  from  that 
of  a  strong  one  (Theorie  me'canique  de  la  chaleur,  1869,  p.  376;  cf.  Plateau,  ii, 
p.  100). 

t  This  identical  phenomenon  was  the  basis  of  Quincke's  theory  of  amoeboid 


280  A  NOTE   ON  ADSORPTION  [ch. 

much  as  to  say  that  the  molecules  of  the  dissolved  or  suspended 
substance  or  substances  will  so  distribute  themselves  throughout 
the  drop  as  to  lead  towards  an  equilibrium,  for  each  small  unit 
of  volume,  between  the  superficial  and  internal  energy ;  or  so,  in 
other  words,  as  to  lead  towards  a  reduction  to  a  minimum  of  the 
potential  energy  of  the  system.  This  tendency  to  concentration 
at  the  surface  of  any  substance  within  the  cell  by  which  the  surface 
tension  tends  to  be  diminished,  or  vice  versa,  constitutes,  then, 
the  phenomenon  of  Adsorption  ;  and  the  general  statement  by 
which  it  is  defined  is  known  as  the  Willard-Gibbs,  or  Gibbs- 
Thomson  law*. 

Among  the  many  important  physical  features  or  concomitants 
of  this  phenomenon,  let  us  take  note  at  present  that  we  need 
not  conceive  of  a  strictly  superficial  distribution  of  the  adsorbed 
substance,  that  is  to  say  of  its  direct  association  with  the  surface 
layer  of  molecules  such  as  we  imagined  in  the  case  of  the  electrical 
charge;  but  rather  of  a  progressive  tendency  to  concentrate, 
more  and  more,  as  the  surface  is  nearly  approached.  Indeed  we 
may  conceive  the  colloid  or  gelatinous  precipitate  in  which,  in  the 
case  of  our  protoplasmic  cell,  the  dissolved  substance  tends  often 
to  be  thrown  down,  to  constitute  one  boundary  layer  after  another, 
the  general  effect  being  intensified  and  multiplied  by  the  repeated 
addition  of  these  new  surfaces. 

Moreover,  it  is  not  less  important  to  observe  that  the  process 
of  adsorption,  in  the  neighbourhood  of  the  surface  of  a  hetero- 
geneous liquid  mass,  is  a  process  which  takes  time ;  the  tendency 
to  surface  concentration  is  a  gradual  and  progressive  one,  and  will 
fluctuate  with  every  minute  change  in  the  composition  of  our 
substance  and  with  every  change  in  the  area  of  its  surface.  In 
other  words,  it  involves  (in  every  heterogeneous  substance)  a 
continual  instability  of  equilibrium  :   and  a  constant  manifestation 

movement  (Ueber  periodische  Ausbreitung  von  Fliissigkeitsoberflachen,  etc.,  SB. 
Berlin.  Akad.  1888,  pp.  791-806;  of.  Pfluger's  Archiv,  1879,  p.  136). 

*  J.  WiUard  Gibbs,  Equilibrium  of  Heterogeneous  Substances,  Tr.  Conn.  Acad. 
Ill,  pp.  380-400,  1876,  also  in  Collected  Papers,  i,  pp.  185-218,  London,  1906; 
J.  J.  Thomson,  Applications  of  Dynamics  to  Physics  and  Chemistry,  1888  (Surface 
tension  of  solutions),  p.  190.  See  also  (int.  al.)  the  various  papers  by  C.  M.  Lewis, 
Phil.  Mag.  (6),  xv,  p.  499,  1908,  xvii,  p.  466,  1909,  ZeUschr.  f.  physik.  Chemie, 
Lxx,  p.  129,  1910;    Milner,  Phil.  Mag.  (6),  xiii,  p.  96,  1907,  etc. 


VI]  THE   GIBBS-THOMSON   LAW  281 

of  motion,  sometimes  in  the  mere  invisible  transfer  of  molecules 
but  often  in  the  production  of  visible  currents  of  fluid  or  manifest 
alterations  in  the  form  or  outUne  of  the  system. 

The  physiologist,  as  we  have  already  remarked,  takes  account 
of  the  general  phenomenon  of  adsorption  in  many  ways :  particu- 
larly in  connection  with  various  results  and  consequences  of 
osmosis,  inasmuch  as  this  process  is  dependent  on  the  presence 
of  a  membrane,  or  membranes,  such  as  the  phenomenon  of  adsorp- 
tion brings  into  existence.  For  instance  it  plays  a  leading  part 
in  all  modern  theories  of  muscular  contraction,  in  which  phenome- 
non a  connection  with  surface  tension  was  first  indicated  by 
E'itzGerald  and  d'Arsonval  nearly  forty  years  ago*.  And,  as 
W.  Ostwald  was  the  first  to  shew,  it  gives  us  an  entirely  new 
conception  of  the  relation  of  gases  (that  is  to  say,  of  oxygen  and 
carbon  dioxide)  to  the  red  corpuscles  of  the  blood  f. 

But  restricting  ourselves,  as  much  as  may  be,  to  our  morpho- 
logical aspect  of  the  case,  there  are  several  ways  in  which  adsorption 
begins  at  once  to  throw  light  upon  our  subject. 

In  the  first  place,  our  prehminary  account,  such  as  it  is,  is 
already  tantamount  to  a  description  of  the  process  of  develop- 
ment of  a  cell-membrane,  or  cell- wall.  The  so-called  "secretion" 
of  this  cell-wall  is  nothing  more  than  a  sort  of  exudation,  or 
striving  towards  the  surface,  of  certain  constituent  molecules  or 
particles  within  the  cell ;  and  the  Gibbs-Thomson  law  formulates, 
in  part  at  least,  the  conditions  under  which  they  do  so.  The 
adsorbed  material  may  range  from  the  almost  unrecognisable 
pelUcle  of  a  blood-corpuscle  to  the  distinctly  differentiated 
''ectosarc"  of  a  protozoan,  and  again  to  the  development  of  a 
fully  formed  cell-wall,  as  in  the  cellulose  partitions  of  a  vegetable 
tissue.  In  such  cases,  the  dissolved  and  adsorbable  material  has 
not  only  the  property  of  lowering  the  surface  tension,  and  hence 

*  G.  F.  FitzGerald,  On  the  Theory  of  Muscular  Contraction,  Brit.  Ass.  RejJ. 
1878;  also  in  Scientific  Writings,  ed.  Larmor,  1902,  pp.  34.  75.  A.  d'Arsonval, 
Relations  entre  I'electricite  animale  et  la  tension  superficielle,  C.  B.  cvi,  p.  1740, 
1888;  cf.  A  Imbert,  Le  mecanisme  de  la  contraction  musculaire,  deduit  de  la  con- 
sideration des  forces  de  tension  superficielle,  Arch,  de  Pliys.  (5),  ix,  pp.  289-301 ,  1897. 

t  Ueber  die  Natur  der  Bindung  der  G^se  im  Blut  und  in  seinen  Bestandtheilen, 
Kolloid.  Zeitschr.  ii,  pp.  264-272,  294-301,  1908;  cf.  Loewy,  Dissociationsspan- 
nung  des  Oxy haemoglobin  ini  Blut,  Arch.  f.  Anat.  und  Physiol.  1904,  p.  231. 


282  A  NOTE   ON  ADSOEPTION  [ch. 

of  itself  accumulating  at  the  surface,  but  has  also  the  property 
of  increasing  the  viscosity  and  mechanical  rigidity  of  the  material 
in  which  it  is  dissolved  or  suspended,  and  so  of  constituting 
a  visible  and  tangible  "membrane*."  The  "zoogloea"  around  a 
group  of  bacteria  is  probably  a  phenomenon  of  the  same  order. 
In  the  superficial  deposition  of  inorganic  materials  we  see  the 
same  process  abundantly  exemplified.  Not  only  do  we  have  the 
simple  case  of  the  building  of  a  shell  or  "test"  upon  the  outward 
surface  of  a  living  cell,  as  for  instance  in  a  Foraminifer.  but  in  a 
subsequent  chapter,  when  we  come  to  deal  with  various  spicules 
and  spicular  skeletons  such  as  those  of  the  sponges  and  of  the 
Radiolaria,  we  shall  see  that  it  is  highly  characteristic  of  the 
whole  process  of  spicule-formation  for  the  deposits  to  be  laid 
down  just  in  the  "  interf acial "  boundaries  between  cells  or 
vacuoles,  and  that  the  form  of  the  spicular  structures  tends  in 
many  cases  to  be  regulated  and  determined  by  the  arrangement 
of  these  boundaries. 

In  physical  chemistry,  an  important  distinction  is  drawn  between  adsorption 
and  pseudo-adsorption'f,  the  former  being  a  reversible,  the  latter  an  irreversible 
or  permanent  phenomenon.  That  is  to  say,  adsorption,  strictly  speaking, 
impUes  the  surface- concentration  of  a  dissolved  substance,  under  circumstances 
which,  if  they  be  altered  or  reversed,  will  cause  the  concentration  to  diminish 
or  disappear.  But  pseudo- adsorption  includes  cases,  doubtless  originating  in 
adsorption  proper,  where  subsequent  changes  leave  the  concentrated  substance 
incapable  of  re-entering  the  liquid  system.  It  is  obvious  that  many  (though 
not  all)  of  our  biological  illustrations,  for  instance  the  formation  of  spicules 
or  of  permanent  cell-membranes,  belong  to  the  class  of  so-called  pseudo- 
adsorption  phenomena.  But  the  apparent  contrast  between  the  two  is  in 
the  main  a  secondary  one,  and  however  important  to  the  chemist  is  of  Uttle 
consequence  to  us. 

*  We  may  trace  the  first  steps  in  the  study  of  this  phenomenon  to  Melsens, 
who  found  that  thin  films  of  white  of  egg  become  firm  and  insoluble  (Sur  les  modi- 
fications apportees  a  ralbumine...par  Taction  purement  mecanique,  C.  R.  Acad. 
Sci.  XXXIII,  p.  247,  1851);  and  Harting  made  similar  observations  about  the  same 
time.  Ramsden  has  investigated  the  same  subject,  and  also  the  more  general 
phenomenon  of  the  formation  of  albuminoid  and  fatty  membranes  by  adsorption : 
cf.  KoaguHerung  der  Eiweisskorper  auf  mechanischer  Wege,  Arch.  f.  Anat.  u.  Phys. 
{Phys.  Ablh.)  1894,  p.  517;  Abscheidung  fester  Korper  in  Oberflachenschichten 
Z.f.  phys.  Chem.  XLVn,  p.  341,  1902;  Proc.  R.  S.  lxxii,  p.  156,  1904.  For  a  general 
review  of  the  whole  subject  see  H.  Zangger,  Ueber  Membranen  und  Membranfunk- 
tionen,  in  Asher-Spiro's  Ergebnisse  der  Physiologic,  vii,  pp.  99-160,  1908. 

t  Cf.  Taylor,  Chemittry  of  Colloids,  p.  252, 


VI]  THE   FORMATION  OF   MEMBRANES  283 

While  this  brief  sketch  of  the  theory  of  membrane-formation 
is  cursory  and  inadequate,  it  is  enough  to  shew  that  the  physical 
theory  of  adsorption  tends  in  part  to  overturn,  in  part  to  simplify 
enormously,  the  older  histological  descriptions.  We  can  no  longer 
be  content  with  such  statements  as  that  of  Strasbiirger,  that 
membrane-formation  in  general  is  associated  with  the  "activity 
of  the  kinoplasm,"  or  that  of  Harper  that  a  certain  spore-membrane 
arises  directly  from  the  astral  rays*.  In  short,  we  have  easily 
reached  the  general  conclusion  that  the  formation  of  a  cell-wall 
or  cell-membrane  is  a  chemico-physical  phenomenon,  which  the 
purely  objective  methods  of  the  biological  microscopist  do  not 
suffice  to  interpret. 

If  the  process  of  adsorption,  on  which  the  formation  of  a 
membrane  depends,  be  itself  dependent  on  the  power  of  the 
adsorbed  substance  to  lower  the  surface  tension,  it  is  obvious  that 
adsorption  can  only  take  place  when  the  surface  tension  already 
present  is  greater  than  zero.  It  is  for  this  reason  that  films  or 
threads  of  creeping  protoplasm  shew  little  tendency,  or  none,  to 
cover  themselves  with  an  encysting  membrane ;  and  that  it  is 
only  when,  in  an  altered  phase,  the  protoplasm  has  developed 
a  positive  surface  tension,  and  has  accordingly  gathered  itself  up 
into  a  more  or  less  spherical  body,  that  the  tendency  to  form  a 
membrane  is  manifested,  and  the  organism  develops  its  "cyst" 
or  cell- wall. 

It  is  found  that  a  rise  of  temperature  greatly  reduces  the 
adsorbability  of  a  substance,  and  this  doubtless  comes,  either  in 
part  or  whole,  from  the  fact  that  a  rise  of  temperature  is  itself 
a  cause  of  the  lowering  of  surface  tension.  We  may  in  all  pro- 
bability ascribe  to  this  fact  and  to  its  converse,  or  at  least  associate 
with  it,  such  phenomena  as  the  encystment  of  unicellular  organisms 
at  the  approach  of  winter,  or  the  frequent  formation  of  strong 
shells  or  membranous  capsules  in  "winter- eggs." 

Again,  since  a  film  or  a  froth  (which  is  a  system  of  films)  can 
only  be  maintained  by  virtue  of  a  certain  viscosity  or  rigidity  of 

*  Strasbiirger,  Ueber  Cytoplasmastrukturen,  etc.  Jahrb.  f.  luiss.  Bot.  xxx> 
1897 ;  R.  A.  Harper,  Kerntheilung  und  freie  Zellbildung  im  Ascus,  ibid. ;  of. 
Wilson,  The  Cell  in  Devdo-pment,  etc.  pp.  53-55. 


284  A  NOTE   ON  ADSORPTION  [ch. 

the  liquid,  it  may  be  quickly  caused  to  disappear  by  the  presence 
in  its  neighbourhood  of  some  substance  capable  of  reducing  the 
surface  tension ;  for  this  substance,  being  adsorbed,  may  displace 
from  the  adsorptive  layer  a  material  to  which  was  due  the  rigidity 
of  the  film.  In  this  way  a  "bathytonic"  substance  such  as  ether 
causes  most  foams  to  subside,  and  the  pouring  oil  on  troubled 
waters  not  only  stills  the  waves  but  still  more  quickly  dissipates 
the  foam  of  the  breakers.  The  process  of  breaking  up  an  alveolar 
network,  such  as  occurs  at  a  certain  stage  in  the  nuclear  division 
of  the  cell,  may  perhaps  be  ascribed  in  part  to  such  a  cause,  as 
well  as  to  the  direct  lowering  of  surface  tension  by  electrical 
agency. 

Our  last  illustration  has  led  us  back  to  the  subject  of  a  previous 
chapter,  namely  to  the  visible  configuration  of  the  interior  of  the 
cell;  and  in  connection  with  this  wide  subject  there  are  many 
phenomena  on  which  light  is  apparently  thrown  by  our  knowledge 
of  adsorption,  and  of  which  we  took  little  or  no  account  in  our 
former  discussion.  One  of  these  phenomena  is  that  visible  or 
concrete  "polarity,"  which  we  have  already  seen  to  be  in  some  way 
associated  with  a  dynamical  polarity  of  the  cell. 

This  morphological  polarity  may  be  of  a  very  simple  kind,  as 
when,  in  an  epithelial  cell,  it  is  manifested  by  the  outward  shape 
of  the  elongated  or  columnar  cell  itself,  by  the  essential  difference 
between  its  free  surface  and  its  attached  base,  or  by  the  presence 
in  the  neighbourhood  of  the  former  of  mucous  or  other  products 
of  the  cell's  activity.  But  in  a  great  many  cases,  this  "polarised" 
symmetry  is  supplemented  by  the  presence  of  various  fibrillae,  or 
of  linear  arrangements  of  particles,  which  in  the  elongated  or 
"monopolar"  cell  run  parallel  with  its  axis,  and  which  tend  to 
a  radial  arrangement  in  the  more  or  less  rounded  or  spherical 
cell.  Of  late  years  especially,  an  immense  importance  has  been 
attached  to  these  various  linear  or  fibrillar  arrangements,  as  they 
occur  {after  staining)  in  the  cell-substance  of  intestinal  epithelium, 
of  spermatocytes,  of  ganglion  cells,  and  most  abundantly  and 
most  frequently  of  all  in  gland  cells.  Various  functions,  which 
seem  somewhat  arbitrarily  chosen,  have  been  assigned,  and  many 
hard  names  given  to  them ;  for  these  structures  now  include  your 
mitochondria  and  your  chondriokonts  (both  of  these  being  varieties 


VI] 


OF   MORPHOLOGICAL   POLARITY 


285 


of  chondriosomes),  your  Altmann's  granules,  your  microsomes, 
pseudo-chromosomes,  epidermal  fibrils  and  basal  filaments,  your 
archeoplasm  and  ergastoplasm,  and  probably  your  idiozomes, 
plasmosomes,  and  many  other  histological  minutiae*. 

The  position  of  these  bodies  with  regard  to  the  other  cell- 
structures  is  carefully  described.  Sometimes  they  lie  in  the 
neighbourhood  of  the  nucleus  itself,  that  is  to  say  in  proximity  to 
the  fluid  boundary  surface  which  separates  the  nucleus  from  the 
cytoplasm ;  and  in  this  position  they  often  form  a  somewhat  cloudy 
sphere  which  constitutes  the  Nebenkern.  In  the  majority  of  cases, 
as  in  the  epithelial  cells,  they  form  filamentous  structures,  and  rows 
.of  granules,  whose  main  direction  is  parallel  to  the  axis  of  the 


IfeiviSi 


A  B  C         . 

Fig.    97.     A,   B,   Cliondriosomes  in   kidney-cells,   prior  to  and  during  secretory 
activity  (after  Barratt);    C,  do.  in  pancreas  of  frog  (after  Mathews). 

cell,  and  which  may,  in  some  cases,  and  in  some  forms,  be  con- 
spicuous at  the  one  end,  and  in  some  cases  at  the  other  end  of 
the  cell.  But  I  do  not  find  that  the  histologists  attempt  to  explain, 
or  to  correlate  with  other  phenomena,  the  tendency  of  these  bodies 
to  lie  parallel  with  the  axis,  and  perpendicular  to  the  extremities 
of  the  cell ;  it  is  merely  noted  as  a  peculiarity,  or  a  specific  character, 
of  these  particular  structures.  Extraordinarily  complicated  and 
diverse  functions  have  been  ascribed  to  them.  Engelmann's 
'■  Fibrillenkonus,"  which  was  almost  certainly  another  aspect  of 
the  same  phenomenon,  was  held  by  him  and  by  cytologists  like 
Breda  and  Heidenhain,  to  be  an  apparatus  connected  in  some 
*  Cf.  A.  Gurwitsch,  Morphologic  und  Biologie  der  Zelle,  1904,  pp.  169-185; 
Meves,  Die  Chondriosomen  als  Trager  erblicher  Anlagen,  Arch.  f.  mikrosk.  Anat. 
1908,  p.  72;  J.  O.  W.  Barratt,  Changes  in  Chondriosomes,  etc.  Q.J. M.S.  LViii, 
pp.  553-566,  1913,  etc. ;  A.  Mathews,  Changes  in  Structure  of  the  Pancreas 
Cell,  etc.,  J.  of  Morph.    xv  (Suppl.),  pp.   171-222,  1899. 


286  A  NOTE   ON   ADSOKPTION  [ch. 

unexplained  way  with  the  mechanism  of  cihary  movement. 
Meves  looked  upon  the  chondriosomes  as  the  actual  carriers  or 
transmitters  of  heredity*.  Altmann  invented  a  new  aphorism, 
Omne  granulum  e  granulo,  as  a  refinement  of  Virchow's  omnis 
cellula  e  cellula ;  and  many  other  histologists,  more  or  less  in  accord, 
accepted  the  chondriosomes  as  important  entities,  sui  generis, 
intermediate  in  grade  between  the  cell  itself  and  its  ultimate 
molecular  components.  The  extreme  cytologists  of  the  Munich 
school,  Popoff,  Goldachmidt  and  others,  following  Kichard  Hertwig, 
declaring  these  structures  to  be  identical  with  "chromidia"  (under 
which  name  Hertwig  ranked  all  extra-nuclear  chromatin),  would 
assign  them  complex  functions  in  maintaining  the  balance  between, 
nuclear  and  cytoplasmic  material ;  and  the  "  chromidial  hypo- 
thesis," as  every  reader  of  recent  cytological  literature  knows,  has 
become  a  very  abstruse  and  complicated  thing  f.  With  the  help 
of  the  " binuclearity  hypothesis''  of  Schaudinn  and  his  school,  it 
has  given  us  the  chromidial  net,  the  chromidial  apparatus,  the 
trophochromidia,  idiochromidia,  gametochroniidia,  the  protogono- 
plasm,  and  many  other  novel  and  original  conceptions.  The 
names  are  apt  to  vary  somewhat  in  significance  from  one  writer 
to  another. 

The  outstanding  fact,  as  it  seems  to  me,  is  that  physiological 
science  has  been  heavily  burdened  in  this  matter,  with  a  jargon 
of  names  and  a  thick  cloud  of  hypotheses ;  while,  from  the  physical 
point  of  view  we  are  tempted  to  see  but  little  mystery  in  the 
whole  phenomenon,  and  to  ascribe  it,  in  all  probability  a'nd  in 
general  terms,  to  the  gathering  or  "clumping"  together,  under 
surface  tension,  of  various  constituents  of  the  heterogeneous  cell- 
content,  and  to  the  drawing  out  of  these  little  clumps  along  the 
axis  of  the  cell  towards  one  or  other  of  its  extremities,  in  relation 
to  osmotic  currents,  as  these  in  turn  are  set  up  in  direct  relation 

*  The  question  whether  chromosomes,  chondriosomes  or  chromidia  be  the  true 
vehicles  or  transmitters  of  "  heredity  "  is  not  without  its  analogy  to  the  older  problem 
of  whether  the  pineal  gland  or  the  pituitary  body  were  the  actual  seat  and  domicile 
of  the  soul. 

f  Of.  C.  C.  Dobell,  Chromidia  and  the  Binuclearity  Hypotheses ;  a  review  and 
a  criticism,  Q.J. M.S.  una,  279-326,  1909;  Prenant,  A.,  Les  Mitochondries  et 
I'Ergastoplasme,  Journ.  de  VAnat.  et  de  la  Physiol.  XLVi,  pp.  217-285,  1910  (both 
with  copious  bibUography). 


VI]  MACALLUM'S   EXPERIMENTS  287 

to  the  phenomena  of  surface  energy  and  of  adsorption*.  And 
all  this  implies  that  the  study  of  these  minute  structures,  if  it 
teach  us  nothing  else,  at  least  surely  and  certainly  reveals  to  us 
the  presence  of  a  definite  "field  of  force,"  and  a  dynamical  polarity 
within  the  cell. 

Our  next  and  last  illustration  of  the  effects  of  adsorption, 

which  we  owe  to  the  investigations  of  Professor  Macallum,  is  of 

great  importance ;    for  it  introduces  us  to  a  series  of  phenomena 

in  regard  to  which  we  seem  now  to  stand  on  firmer  ground  than 

in  some  of  the  foregoing  cases,  though  we  cannot  yet  consider  that 

the  whole  story  has  been  told.     In  our  last  chapter  we  were 

restricted  mainly,  though  not  entirely,  to  a  consideration  of  figures 

of  equilibrium,  such  as  the  sphere,  the  cylinder  or  the  unduloid ; 

and  we  began  at  once  to  find  ourselves  in  difficulties  when  we  were 

confronted  by  departures  from  symmetry,  as  for  instance  in  the 

simple  case  of  the  ellipsoidal  yeast-cell  and  the  production  of  its 

bud.     We  found  the  cylindrical  cell  of  Spirogyra,  with  its  plane 

or  spherical  ends,  a  comparatively  simple  matter  to  understand; 

but  when  this  uniform  cylinder  puts  out  a  lateral  outgrowth,  in 

the  act  of  conjugation,  we  have  a  new  and  very  different  system 

of  forces  to  explain.     The  analogy  of  the  soap-bubble,  or  of  the 

simple  liquid  drop,  was  apt  to  lead  us  to  suppose  that  the  surface 

tension  ivas,  on  the  whole,  uniform  over  the  surface  of  our  cell; 

and  that  its  departures  from  symmetry  of  form  were  therefore 

likely  to  be  due  to  variations  in  external  resistance.     But  if  we 

have  been  inclined  to  make  such  an  assumption  we  must  now 

*  Traube  in  particular  has  maintained  that  in  differences  of  surface-tension 
we  have  the  origin  of  the  active  force  productive  of  osmotic  currents,  and  that 
herein  we  find  an  explanation,  or  an  approach  to  an  explanation,  of  many  phenomena 
which  were  formerly  deemed  peculiarly  "vital"  in  their  character.  "Die  Differenz 
der  Oberflachenspannungen  oder  der  Oberflachendruck  eine  Kraft  darsteUt,  welche 
als  treibende  Kraft  der  Osmose,  an  die  Stelle  des  nicht  mit  dem  Oberflachendruck 
identischen  osmotischen  Druckes,  zu  setzen  ist,  etc."  (Oberflachendi-uck  und 
•seine  Bedeutung  im  Organismus,  Pfliiger's  Archiv,  cv,  p.  559,  1904.)  Cf.  also 
Hardy  {Pr.  Phys.  Soc.  xxvm,  p.  116,  1916),  "If  the  surface  film  of  a  colloid 
membrane  separating  two  masses  of  fluid  were  to  change  in  such  a  way  as  to  lower 
the  potential  of  the  water  in  it,  water  would  enter  the  region  from  both  sides  at 
once.  But  if  the  change  of  state  were  to  be  propagated  as  a  wave  of  change, 
starting  at  one  face  and  dying  out  at  the  other  face,  water  would  be  carried  along" 
from  one  side  of  the  membrane  to  the  other.  A  succession  of  such  waves  would 
maintain  a  flow  of  fluid." 


288  A  NOTE   ON  ADSORPTION  [ch. 

reconsider  it,  and  be  prepared  to  deal  with  important  localised 
variations  in  the  surface  tension  of  the  cell.  For,  as  a  matter  of 
fact,  the  simple  case  of  a  perfectly  symmetrical  drop,  with  uniform 
surface,  at  which  adsorption  takes  place  with  similar  uniformity, 
is  probably  rare  in  physics,  and  rarer  still  (if  it  exist  at  all)  in  the 
fluid  or  fluid-containing  system  which  we  call  in  biology  a  cell. 
We  have  mostly  to  do  with  cells  whose  general  heterogeneity  of 
substance  leads  to  qualitative  differences  of  surface,  and  hence  to 
varying  distributions  of  surface  tension.  We  must  accordingly 
investigate  the  case  of  a  cell  which  displays  some  definite  and 
regular  heterogeneity  of  its  liquid  surface,  just  as  Amoeba  displays 
a  heterogeneity  w^hich  is  complex,  irregular  and  continually 
fluctuating  in  amount  and  distribution.  Such  heterogeneity  as 
we  are  speaking  of  must  be  essentially  chemical,  and  the  prelimin- 
ary problem  is  to  devise  methods  of  "microchemical"  analysis, 
which  shall  reveal  localised  accumulations  of  particular  substances 
within  the  narrow  limits  of  a  cell,  in  the  hope  that,  their  normal 
effect  on  surface  tension  being  ascertained,  we  may  then  correlate 
with  their  presence  and  distribution  the  actual  indications  of 
varying  surface  tension  which  the  form  or  movement  of  the  cell 
displays.  In  theory  the  method  is  all  that  we  could  wish,  but  in 
practice  we  must  be  content  with  a  very  limited  application  of  it ; 
for  the  substances  which  may  have  such  action  as  we  are  looking 
for.  and  which  are  also  actual  ^r  possible  constituents  of  the  cell, 
are  very  numerous,  while  the  means  are  very  seldom  at  hand  to 
demonstrate  their  precise  distribution  and  localisation.  But  in 
one  or  two  cases  we  have  such  means,  and  the  most  notable  is  in 
connection  with  the  element  potassium.  As  Professor  Macallum 
has  shewn,  this  element  can  be  revealed,  in  very  minute  quantities, 
by  means  of  a  certain  salt,  a  nitrite  of  cobalt  and  sodium*.  This 
salt  penetrates  readily  into  the  tissues  and  into  the  interior  of  the 
cell;  it  combines  with  potassium  to  form  a  sparingly  soluble 
nitrite  of  cobalt,  sodium  and  potassium ;  and  this,  on  subsequent 
treatment  with  ammonium  sulphide,  is  converted  into  a  character- 
istic black  precipitate  of  cobaltic  sulphide  f. 

*  On  the  Distribution  of  Potassium  in  animal  and  vegetable  Cells;   Journ.  of 
Physiol.  XXXII,  p.  95,  1905. 

f  The  reader  will  recognise  that  there  is  a  fundamental  difference,  and  contrast. 


VI]  MACALLUM'S   EXPERIMENTS  289 

By  this  means  Macalluni  demonstrated  some  years  ago  the 
unexpected  presence  of  accumulations  of  potassium  (i.e.  of  chloride 
or  other  salts  of  potassium)  localised  in  particular  parts  of  various 
cells,  both  solitary  cells  and  tissue  cells ;  and  he  arrived  at  the 
conclusion  that  the  localised  accumulations  in  question  were 
simply  evidences  of  concentration  of  the  dissolved  potassium  salts, 
formed  and  localised  in  accordance  with  the  Gibbs-Thomson  law. 
In  other  words,  these  accumulations,  occurring  as  they  actually  do 
in  connection  with  various  boundary  surfaces,  are  evidence,  when 
they  appear  irregularly  distributed  over  such  a  surface,  of  in- 
equalities in  its  surface  tension* ;  and  we  may  safely  take  it  that 
our  potassium  salts,  like  inorganic  substances  in  general,  tend  to 
raise  the  surface  tension,  and  will  therefore  be  found  concentrating 
at  a  portion  of  the  surface  whose  tension  is  weakf. 

In  Professor  Macallum's  figure  (Fig.  98,  1)  of  the  little  green 
alga  Pleurocarpus,  we  see  that  one  side  of  the  cell  is  beginning  to 
bulge  out  ill  a  wide  convexity.  This  bulge  is,  in  the  first  place, 
a  sign  of  weakened  surface  tension  on  one  side  of  the  cell,  which  as 
a  whole  had  hitherto  been  a  symmetrical  cyhnder ;  in  the  second 
place,  we  see  that  the  bulging  area  corresponds  to  the  position  of 
a  great  concentration  of  the  potassic  salt ;  while  in  the  third  place, 
from  the  physiological  point  of  view,  we  call  the  phenomenon 
the  first  stage  in  the  process  of  conjugation.  In  Fig.  98,  2,  of 
Mesocarpus  (a  close  ally  of  Spirogyra),  we  see  the  same  phenomenon 
admirably  exemplified  in  a  later  stage.  From  the  adjacent  cells 
distinct  outgrowths  are  being  emitted,  where  the  surface  tension  has 
been  weakened :  just  as  the  glass-blower  warms  and  softens  a  small 
part  of  his  tube  to  blow  out  the  softened  area  into  a  bubble  or 
diverticulum ;  and  in  our  Mesocarpus  cells  (besides  a  certain 
amount  of  potassium  rendered  visible  over  the  boundary  which 

between  such  experiments  as  these  of  Professor  Macallum's  and  the  ordinary 
staining  processes  of  the  histologist.  The  latter  are  (as  a  general  rule)  purely- 
empirical,  while  the  former  endeavour  to  reveal  the  true  microchemistry  of  the 
cell.  "On  pent  dire  que  la  microchimie  n'est  encore  qu'a  la  periode  d'essai,  et 
que  I'avenir  de  I'histologie  et  specialement  de  la  cytologie  est  tout  entier  dans  la 
microchimie"  (Prenant,  A.,  Methodes  et  resultats  de  la  Microchimie,  Jourti.  de 
VAnat.  et  de  la  Physiol,  xlvi,  pp.  343-404,  1910). 

*  Cf.  Macallum,  Presidential  Address,  Section  I,  Brit.  Ass.  Rep.  (Sheffield), 
1910,  p.  744. 

t  In  accordance  with  a  simple  corollary  to  the  Gibbs-Thomson  law. 

T.  G.  19 


290 


A  NOTE   OF   ADSORPTION 


[CH. 


separates  the  green  protoplasm  from  the  cell-sap),  there  is  a  very 
large  accumulation  precisely  at  the  point  where  the  tension  of  the 
originally  cyUndrical  cell  is  weakening  to  produce  the  bulge. 
But  in  a  still  later  stage,  when  the  boundary  between  the  two 
conjugating  cells  is  lost  and  the  cytoplasm  of  the  two  cells  becomes 
fused  together,  then  the  signs  of  potassic  concentration  quickly 
disappear,  the  salt  becoming  generally  diffused  through  the  now 
symmetrical  and  spherical  "zygospore." 


i^^^ 


Fig.  98.  Adsorptive  concentration  of  potassium  salts  in  (1)  cell  of  Pleurocarpus 
about  to  conjugate;  (2)  conjugating  cells  of  Mesocarpus;  (3)  sprouting  spores 
oi  Equisetum.     (After  Macallum.) 


In  a  spore  of  Equisetum  (Fig.  98,  3),  while  it  is  still  a  single  cell, 
no  localised  concentration  of  potassium  is  to  be  discerned ;  but  as 
soon  as  the  spore  has  divided,  by  an  internal  partition,  into  two 
cells,  the  potassium  salt  is  found  to  be  concentrated  in  the  smaller 
one,  and  especially  towards  its  outer  wall,  which  is  marked  by  a 
pronounced  convexity.  And  as  this  convexity  (which  corresponds 
to  one  pole  of  the  now  asymmetrical,  or  quasi-ellipsoidal  spore) 
grows  out  into  the  root-hair,  the  potassium  salt  accompanies  its 
growth,  and  is  concentrated  under  its  wall.     The  concentration  is. 


VI]  MACALLUM'S   EXPERIMENTS  291 

accordingly,  a  concomitant  of  the  diminished  surface  tension  which 
is  manifested  in  the  altered  configuration  of  the  system. 

In  the  case  of  ciliate  or  flagellate  cells,  there  is  to  be  found  a 
characteristic  accumulation  of  potassium  at  and  near  the  base  of 
the  cilia.  The  relation  of  ciliary  movement  to  surface  tension 
hes  beyond  our  range,  but  the  fact  which  we  have  just  mentioned 
throws  hght  upon  the  frequent  or  general  presence  of  a  little 
protuberance  of  the  cell-surface  just  where  a  flagellum  is  given 
of£  (cf.  p.  247),  and  of  a  little  projecting  ridge  or  fillet  at  the  base 
of  an  isolated  row  of  cilia,  such  as  we  find  in  Vorticella. 

Yet  another  of  Professor  Macallum's  demonstrations,  though 
its  interest  is  mainly  physiological,  will  help  us  somewhat  further 
to  comprehend  what  is  implied  in  our  phenomenon.  In  a  normal 
cell  of  Spirogyra,  a  concentration  of  potassium  is  revealed  along 
the  whole  surface  of  the  spiral  coil  of  chlorophyll-bearing,  or 
"  chromatophoral,"  protoplasm,  the  rest  of  the  cell  being  wholly 
destitute  of  the  former  substance :  the  indication  being  that,  at 
this  particular  boundary,  between  chromatophore  and  cell-sap, 
the  surface  tension  is  small  in  comparison  with  any  other  interfacial 
surface  within  the  system. 

Now  as  Macallum  points  out,  the  presence  of  potassium  is 
known  to  be  a  factor,  in  connection  with  the  chlorophyll-bearing 
protoplasm,  in  the  synthetic  production  of  starch  from  COg  under 
the  influence  of  sunlight.  But  we  are  left  in  some  doubt  as  to 
the  consecutive  order  of  the  phenomena.  For  the  lowered  surface 
tension,  indicated  by  the  presence  of  the  potassium,  may  be 
itself  a  cause  of  the  carbohydrate  synthesis ;  while  on  the  other 
hand,  this  synthesis  may  be  attended  by  the  production  of  sub- 
stances (e.g.  formaldehyde)  which  lower  the  surface  tension,  and 
so  conduce  to  the  concentration  of  potassium.  All  we  know  for 
certain  is  that  the  several  phenomena  are  associated  with  one 
another,  as  apparently  inseparable  parts  or  ine\'itable  concomitants 
of  a  certain  complex  action. 

And  now  to  return,  for  a  moment,  to  the  question  of  cell-form. 
When  we  assert  that  the  form  of  a  cell  (in  the  absence  of  mechanical 
pressure)  is  essentially  dependent  on  surface  tension,  and  even  when 
we  make  the  preliminary  assumption  that  protoplasm  is  essentially 

19—2 


292  A  NOTE   OF   ADSORPTION  [ch.  vi 

a  fluid,  we  are  resting  our  belief  on  a  general  consensus  of  evidence, 
rather  than  on  compliance  with  any  one  crucial  definition.  The 
simple  fact  is  that  the  agreement  of  cell-forms  with  the  forms 
which  physical  experiment  and  mathematical  theory  assign  to 
liquids  under  the  influence  of  surface  tension,  is  so  frequently  and 
often  so  typically  manifested,  that  we  are  led,  or  driven,  to  accept 
the  surface  tension  hypothesis  as  generally  applicable  and  as 
eqmvaient  to  a  universal  law.  The  occasional  difficulties  or 
apparent  exceptions  are  such  as  call  for  further  enquiry,  but  fall 
short  of  throwing  doubt  upon  our  hypothesis.  Macallum's 
researches  introduce  a  new  element  of  certainty,  a  "nail  in  a  sure 
place,"  wnen  tliey  demonstrate  that,  in  certain  movements  or 
changes  of  form  which  we  should  naturally  attribute  to  weakened 
surface  tension,  a  chemical  concentration  which  would  naturally 
accompany  such  weakening  actually  takes  place.  They  further 
teach  us  that  in  the  cell  a  chemical  heterogeneity  may  exist  of 
a  very  marked  kind,  certain  substances  being  accumulated  here 
and  absent  there,  within  the  narrow  bounds  of  the  system. 

Such  localised  accumulations  can  as  yet  only  be  demonstrated 
in  the  case  of  a  very  few  substances,  and  of  a  single  one  in  par- 
ticular ;  and  these  are  substances  whose  presence  does  not  produce, 
but  whose  concentration  tends  to  follow,  a  weakening  of  surface 
tension.  The  physical  cause  of  the  localised  inequalities  of  surface 
tension  remains  unknown.  We  may  assume,  if  we  please,  that  it 
is  due  to  the  prior  accumulation,  or  local  production,  of  chemical 
bodies  which  would  have  this  direct  effect;  though  we  are  by 
no  means  limited  to  this  hypothesis. 

But  in  spite  of  some  remaining  difficulties  and  uncertainties, 
we  have  arrived  at  the  conclusion,  as  regards  unicellular  organisms, 
that  not  only  their  general  configuration  but  also  their  departures 
from  symmetry  may  be  correlated  with  the  molecular  forces 
manifested  in  their  fluid  or  semi-fluid  surfaces. 


CHAPTER    VII 

THE   FORMS   OF  TISSUES   OR   CELL-AGGREGATES 

We  now  pass  from  the  consideration  of  the  solitary  cell  to  that 
of  cells  in  contact  with  one  another, — to  what  we  may  call  in 
the  first  instance  "  cell-aggregates," — through  which  we  shall  be  led 
ultimately  to  the  study  of  complex  tissues.  In  this  part  of  our 
subject,  as  in  the  preceding  chapters,  we  shall  have  to  give  some 
consideration  to  the  efEects  of  various  forces ;  but,  as  in  the  case 
of  the  conformation  of  the  solitary  cell,  we  shall  probably  find, 
and  we  may  at  least  begin  by  assuming,  that  the  agency  of  surface 
tension  is  especially  manifest  and  important.  The  effect  of  this 
surface  tension  will  chiefly  manifest  itself  in  the  production  of 
surfaces  mmimae  areae :  where,  as  Plateau  was  always  careful  to 
point  out,  we  must  understand  by  this  expression  not  an  absolute, 
but  a  relative  minimum,  an  area,  that  is  to  say,  which  approxi- 
mates to  an  absolute  minimum  as  nearly  as  circumstances  and  the 
conditions  of  the  case  permit. 

There  are  certain  fundamental  principles,  or  fundamental 
•equations,  besides  those  which  we  have  already  considered,  which 
we  shall  need  in  our  enquiry.  For  instance  the  case  which  we 
briefly  touched  upon  (on  p.  265)  of  the  angle  of  contact  between 
the  protoplasm  and  the  axial  filament  in  a  Heliozoan  we  shall 
now  find  to  be  but  a  particular  case  of  a  general  and  elementary 
theorem. 

Let  us  re-state  as  follows,  in  terms  of  Energij,  the  general 
principle  which  underlies  the  theory  of  surface  tension  or  capillarity. 

When  a  fluid  is  in  contact  with  another  fluid,  or  with  a  solid 
or  a  gas,  a  portion  of  the  energy  of  the  system  (that,  namely, 
which  we  call  surface  energy),  is  proportional  to  the  area  of  the 
surface  of  contact :  it  is  also  proportional  to  a  coefficient  which 
is  specific  for  each  particular  pair  of  substances,  and  which  is 
constant  for  these,  save  only  in  so  far  as  it  may  be  modified  by 


294  THE  FORMS   OF   TISSUES  [ch. 

changes  of  temperature  or  of  electric  charge.  The  condition  of 
minimum  potential  energy  in  the  system,  which  is  the  condition  of 
equihbrium,  will  accordingly  be  obtained  by  the  utmost  possible 
diminution  in  the  area  of  the  surfaces  in  contact.  When  we  have 
three  bodies  in  contact,  the  case  becomes  a  little  more  complex. 
Suppose  for  instance  we  have  a  drop  of  some  fluid.  A,  floating  on 
another  fluid,  B,  and  exposed  to  air,  C.  The  whole  surface  energy 
of  the  system  may  now  be  considered  as  divided  into  two  parts, 
one  at  the  surface  of  the  drop,  and  the  other  outside  of  the  same ; 
the  latter  portion  is  inherent  in  the  surface  BC,  between  the  mass 
of  fluid  B  and  the  superincumbent  air,  C ;  but  the  former  portion 
consists  of  two  parts,  for  it  is  divided  between  the  two  surfaces  AB 
and  AC,  that  namely  which  separates  the  drop  from  the  surrounding 
fluid  and  that  which  separates  it  from  the  atmosphere.     So  far  as 


the  drop  is  concerned,  then,  equihbrium  depends  on  a  proper 
balance  between  the  energy,  per  unit  area,  which  is  resident  in 
its  own  two  surfaces,  and  that  which  is  external  thereto :  that  is 
to  say,  if  we  call  Ejjf.  the  energy  at  the  surface  between  the  two 
fluids,  and  so  on  with  the  other  two  pairs  of  surface  energies,  the 
condition  of  equilibrium,  or  of  maintenance  of  the  drop,  is  that 

If,  on  the  other  hand,  the  fluid  A  happens  to  be  oil  and  the  fluid 
B,  water,  then  the  energy  per  unit  area  of  the  water-air  surface 
is  greater  than  that  of  the  oil-air  surface  and  that  of  the  oil-water 
surface  together ;  i.e. 

E     >  E    +  E 

Here  there  is  no  equilibrium,  and  in  order  to  obtain  it  the  water-air 
surface  must  always  tend  to  decrease  and  the  other  two  interfacial 
surfaces  to  increase ;  which  is  as  much  as  to  say  that  the  water 
tends  to  become  covered  by  a  spreading  film  of  oil,  and  the  water- 
air  surface  to  be  abolished. 


VII]  OK   CELL-AGGREGATES  295 

The  surface  energy  of  which  we  have  here  spoken  is  manifested 
in  that  contractile  force,  or  "tension,"  of  which  we  have  already 
had  so  much  to  say*.  In  any  part  of  the  free  water  surface,  for 
instance,  one  surface  particle  attracts  another  surface  particle,  and 
the  resultant  of  these  multitudinous  attractions  is  an  equilibrium 
of  tension  throughout  this  particular  surface.  In  the  case  of  our 
three  bodies  in  contact  with  one  another,  and  within  a  small  area 
very  near  to  the  point  of  contact,  a  water  particle  (for  instance) 
will  be  pulled  outwards  by  another  water  particle;  but  on  the 
opposite  side,  so  to  speak,  there  will  be  no  water  surface,  and  no 
water  particle,  to  furnish  the  counterbalancing  pull ;  this  counter- 


Fig.  100. 


Fig.   101. 


pull,  which  is  necessary  for  equilibrium,  must  therefore  be  provided 
by  the  tensions  existing  in  the  other  two  surfaces  of  contact.  In 
short,  if  we  could  imagine  a  single  particle  placed  at  the  very  point 
of  contact,  it  would  be  drawn  upon  by  three  different  forces, 
whose  directions  would  lie  in  the  three  surface  planes,  and  whose 
magnitude  would  be  proportional  to  the  specific  tensions  charac- 
teristic of  the  two  bodies  which  in  each  case  combine  to  form  the 
"interfacial"  surface.  Now  for  three  forces  acting  at  a  point  to 
be  in  equilibrium,  they  must  be  capable  of  representation,  in 
magnitude  and  direction,  by  the  three  sides  of  a  triangle,  taken  in 
order,  in  accordance  with  the  elementary  theorem  of  the  Triangle 
of  Forces.  So,  if  we  know  the  form  of  our  floating  drop  (Fig.  100), 
then  by  drawing  tangents  from  0  (the  point  of  mutual  contact), 

*  It  can  easily  be  proved  (by  equating  the  increase  of  energy  stored  in  an 
increased  surface  to  the  work  done  \.\  increasing  that  surface),  that  the  tension 
measured  per  unit  breadth,  T„,,,  is  equal  to  the  energy  per  unit  area,  ^n,,. 


296  THE   FOEMS   OF   TISSUES  [ch. 

we  determine  the  three  angles  of  our  triangle  (Fig.  101),  and  we 
therefore  know  the  relative  magnitudes  of  the  three  surface 
tensions,  which  magnitudes  are  proportional  to  its  sides;  and 
conversely,  if  we  know  the  magnitudes,  or  relative  magnitudes, 
of  the  three  sides  of  the  triangle,  we  also  know  its  angles,  and  these 
determine  the  form  of  the  section  of  the  drop.  It  is  scarcely- 
necessary  to  mention  that,  since  all  points  on  the  edge  of  the 
drop  are  under  similar  conditions,  one  with  another,  the  form  of 
the  drop,  as  we  look  down  upon  it  from  above,  must  be  circular, 
and  the  whole  drop  must  be  a  solid  of  revolution. 

The  principle  of  the  Triangle  of  Forces  is  expanded,  as  follows, 
by  an  old  seventeenth-century  theorem,  called  Lami's  Theorem : 
"If  three  forces  acting  at  a  -point  he  in  equilihrium,  each  force  is 
-proportional  to  the  sine  of  the  angle  contained  between  the  directions 
of  the  other  two.''^     That  is  to  say 

P:Q:R:  =  sm  QOR  :  sin  FOR  :  sin  POQ, 

P Q        _       R^ 

®^  sin  QOR  ~  sin  ROP  ~  sin  POQ  ' 

And  from  this,  in  turn,  we  derive  the  equivalent  formulae,  by 
which  each  force  is  expressed  in  terms  of  the  other  two,  and  of  the 
angle  between  them : 

P2  =  ^2  +  2^2  +  2QR  cos  [QOR),  etc. 

From  this  and  the  foregoing,  we  learn  the  following  important 
and  useful  deductions : 

(1)  The  three  forces  can  only  be  in  equilibrium  when  any  one 
of  them  is  less  than  the  sum  of  the  other  two :  for  otherwise,  the 
triangle  is  impossible.  Now  in  the  case  of  a  drop  of  olive-oil 
upon  a  clean  water  surface,  the  relative  magnitudes  of  the  three 
tensions  (at  15°  C.)  have  been  determined  as  follows: 

Water-air  surface  ...  ...         75 

Oil-air  surface    ...  ...  ...         32 

Oil-water  surface  ...         ...         21 

No  triangle  having  sides  of  these  relative  magnitudes  is  possible ; 
and  no  such  drop  therefore  can  remain  in  equilibrium. 


VII]  OF  SACHS'S   RULE  297 

(2)  The  three  surfaces  may  be  all  alike:  as  when  a  soap- 
bubble  floats  upon  soapy  water,  or  when  two  soap-bubbles  are 
joined  together,  on  either  side  of  a  partition-film.  In  this  case, 
the  three  tensions  are  all  equal,  and  therefore  the  three  angles 
are  all  equal ;  that  is  to  say,  when  three  similar  liquid  surfaces 
meet  together,  they  always  do  so  at  an  angle  of  120°.  Whether 
our  two  conjoined  soap-bubbles  be  equal  or  unequal,  this  is  still 
the  invariable  rule ;  because  the  specific  tension  of  a  particular 
surface  is  unaffected  by  any  changes  of  magnitude  or  form. 

(3)  If  two  only  of  the  surfaces  be  ahke,  then  two  of  the 
angles  will  be  alike,  and  the  other  will  be  unlike;  and  this  last 
will  be  the  difference  between  360°  and  the  sum  of  the  other  two. 
A  particular  case  is  when  a  film  is  stretched  between  solid  and 
parallel  walls,  like  a  soap-film  within  a  cylindrical  tube.  Here,  so 
long  as  there  is  no  external  pressure  applied  to  either  side,  so  long 
as  both  ends  of  the  tube  are  open  or  closed,  the  angles  on  either 
side  of  the  film  will  be  equal,  that  is  to  say  the  film  will  set  itself 
at  right  angles  to  the  sides. 

Many  years  ago  Sachs  laid  it  down  as  a  principle,  which  has 
become  celebrated  in  botany  under  the  name  of  Sachs's  Rule, 
that  one  cell- wall  always  tends  to  set  itself  at  right  angles  to  another 
cell-wall.  This  rule  applies  to  the  case  which  we  have  just  illus- 
trated; and  such  validity  as  the  rule  possesses  is  due  to  the  fact 
that  among  plant-tissues  it  very  frequently  happens  that  one 
cell-wall  has  become  solid  and  rigid  before  another  and  later 
partition-wall  is  developed  in  connection  with  it. 

(4)  There  is  another  important  principle  which  arises  not  out 
of  our  equations  but  out  of  the  general  considerations  by  which 
we  were  led  to  them.  We  have  seen  that,  at  and  near  the  point 
of  contact  between  our  several  surfaces,  there  is  a  continued 
balance  of  forces,  carried,  so  to  speak,  across  the  interval;  in 
other  words,  there  is  physical  continuity  between  one  surface  and 
another.  It  follows  necessarily  from  this  that  the  surfaces  merge 
one  iiito  another  by  a  continuous  curve.  Whatever  be  the  form 
of  our  surfaces  and  whatever  the  angle  between  them,  this  small 
intervening  surface,  approximately  spherical,  is  always  there  to 
bridge  over  the  line  of  contact*  ;  and  this  little  fillet,  or  "  bourrelet," 

*  The  presence  of  this  httle  liquid  '"bourrelet,"  drawn  from  tlie  material  of  which 


298 


THE  FORMS   OF  TISSUES 


[CH. 


as  Plateau  called  it,  is  large  enough  to  be  a  common  and  con- 
spicuous feature  in  the  microscopy  of  tissues  (Fig.  102).  For 
instance,  the  so-called  "splitting"  of  the  cell- wall,  which  is  con- 
spicuous at  the  angles  of  the  large  "parenchymatous"  cells  in  the 
succulent  tissues  of  all  higher  plants  (Fig.  103),  is  nothing  more 
than  a  manifestation  of  Plateau's  "bourrelet,"  or  surface  of 
continuity*. 

We  may  now  illustrate  some  of  the  foregoing  principles., 
before  we  proceed  to  the  more  complex  cases  in  which  more 
bodies  than  three  are  in  mutual  contact.  But  in  doing  so,  we 
must  constantly  bear  in  mind  the  principles  set  forth  in  our 
chapter  on  the  forms  of  cells,  and  especially  those  relating  to  the 
pressure  exercised  by  a  purved  film. 


Fig.  102.     (After  Berthold.) 


Fig.  103.     Parenchyma  of  Maize. 


Let  us  look  for  a  moment  at  the  case  presented  by  the  partition- 
wall  in  a  double  soap-bubble.  As  we  have  just  seen,  the  three 
films  in  contact  (viz.  the  outer  walls  of  the  two  bubbles  and  the 
partition-wall  between)  being  all  composed  of  the  same  substance 

the  partition -walls  themselves  are  composed,  is  obviously  tending  to  a  reduction 
of  the  internal  surface-area.  And  it  may  be  that  it  is  as  well,  or  better,  accounted 
for  on  this  ground  than  on  Plateau's  assumption  that  it  represents  a  "surface  of 
continuity." 

*  A  similar  "bourrelet"  is  admirably  seen  at  the  line  of  junction  between  a 
floating  bubble  and  the  liquid  on  which  it  floats;  in  which  case  it  constitutes  a 
"masse  annulaire,"  whose  mathematical  properties  and  relation  to  the  form  of  the 
nearly  hemispherical  bubble,  have  been  investigated  by  van  der  Mensbrugghe  {cf. 
Plateau,  oj).  cit.,  p.  386).  The  form  of  the  superficial  vacuoles  in  Actinophrys  or 
Actinosphaerium  involves  an  identical  problem. 


VIl] 


OF  PLATEAU'S   BOURRELET 


299 


and  all  alike  iu  contact  with  air,  the  three  surface  tensions  must 
be  equal ;  and  the  three  films  must  therefore,  in  all  cases,  meet 
at  an  angle  of  120°.  But,  unless  the  two  bubbles  be  of  precisely 
equal  size  (and  therefore  of  equal  curvature)  it  is  obvious  that  the 
tangents  to  the  spheres  will  not  meet  the  plane  of  their  circle 
of  contact  at  equal  angles,  and  therefore  that  the  partition-wall 
must  be  a  curved  surface :  it  is  only  plane  when  it  divides  two 
equal  and  symmetrical  cells.  It  is  also  obvious,  from  the  sym- 
metry of  the  figure,  that  the  centres  of  the  spheres,  the  centre  of 
the  partition,  and  the  centres  of  the  two  spherical  surfaces  are 
all  on  one  and  the  same  straight  fine. 

Now  the  surfaces  of  the  two  bubbles  exert  a  pressure  inwards 
which  is  inversely  proportional  to  their  radii :  that  is  to  say 
])  :  J)'  :  :  l/r'  :  1/r ;  and  the  partition  wall  must,  for  equilibrium, 
exert  a  pressure  (P)  which  is  equal  to  the  difference  between  these 


Fig.  104. 

two  pressures,  that  is  to  say,  P  =  l/R  =  l/r'  —  l/r  =  (r  —  r')/rr'.  It 
follows  that  the  curvature  of  the  partition  wall  must  be  just  such 
a  curvature  as  is  capable  of  exerting  this  pressure,  that  is  to  say, 
R  =  rr'/{r  —  r').  The  partition  wall,  then,  is  always  a  portion  of 
a  spherical  surface,  whose  radius  is  equal  to  the  product,  divided 
by  the  difference,  of  the  radii  of  the  two  vesicles.  It  follows  at 
once  from  this  that  if  the  two  bubbles  be  equal,  the  radius  of 
curvature  of  the  partition  is  infinitely  great,  that  is  to  say  the 
partition  is  (as  we  have  already  seen)  a  plane  surface. 

The  geometrical  construction  by  which  we  obtain  the  position 
of  the  centres  of  the  two  spheres  and  also  of  the  partition  surface 
is  very  simple,  always  provided  that  the  surface  tensions  are 
uniform  throughout  the  system.  If  ^  be  a  point  of  contact 
between  the  two  spheres,  and  cp  be  a  radius  of  one  of  them,  then 
make  the  angle  cpm  =  60°,  and  mark  off  on  pm,  pc'  equal  to  the 


300 


THE  FORMS   OF   TISSUES 


[CH. 


radius  of  the  other  sphere ;  in  Uke  manner,  make  the  angle 
c'jyn  =  60°,  cutting  the  Hne  cc'  in  c" ;  then  c'  will  be  the  centre 
of  the  second  sphere,  and  c"  that  of  the  spherical  partition. 

Whether  the  partition  be  or  be  not  a  plane  surface,  it  is  obvious 
that  its  line  of  junction  with  the  rest  of  the  system  lies  in  a  plane. 


Fig.  105. 


Fig.  106. 


and  is  at  right  angles  to  the  axis  of  symmetry.  The  actual 
curvature  of  the  partition- wall  is  easily  seen  in  optical  section ; 
but  in  surface  view,  the  line  of  junction  is  projected  as  a  plane 
(Fig.  106),  perpendicular  to  the  axis,  and  this  appearance  has 
also  helped  to  lend  support  and  authority  to  "  Sachs's  Rule." 


B 


oooocCJcco 
ooooooo  ; 

OOCZ3CIDO. 


Many  spRerical  cells,  such  as 
Protococcus,  divide  into  two  equal 
halves,  which  are  therefore  separ- 
ated by  a  plane  partition.  Among 
the  other  lower  Algae,  akin  to 
Protococcus,  such  as  the  Nostocs 
and  Oscillatoriae,  in  which  the 
cells  are  imbedded  in  a  gelatinous 
matrix,  we  find  a  series  of  forms 
such  as  are  represented  in  Fig.  107. 
Sometimes  the  cells  are  solitary 
or  disunited;  sometimes  they  run 
in  pairs  or  in  rows,  separated  one 
from  another  by  flat  partitions ; 
and  sometimes  the  conjoined  cells 
are  approximately  hemispherical,  but  at  other  times  each  half 
is  more  than   a   hemisphere.     These  various  conditions  depend, 


c  QGoxECDaio: 


Fig.  107.  Filaments,  or  chains  of 
cells,  in  various  lower  Algae. 
(A)  NoHoc;  (B)  Anabaena;  (C) 
Rivularia;    (D)  Oscillatorin. 


VIl] 


OF  CELL-PARTITIONS 


301 


according  to  what  we  have  already  learned,  upon  the  relative 
magnitudes  of  the  tensions  at  the  surface  of  the  cells  and  at  the 
boundary  between  them*. 

In  the  typical  case  of  an  equally  divided  cell,  such  as  a  double 
and  co-equal  soap-bubble,  where  the  partition-wall  and  the  outer 
walls  are  similar  to  one  another  and  in  contact  with  similar  sub- 
stances, we  can  easily  determine  the  form  of  the  system.  For,  at 
any  point  of  the  boundary  of  the  partition-wall,  0.  the  tensions 
being  equal,  the  angles  QOP,  ROP,  QOR  are  all  equal,  and  each 
is,  therefore,  an  angle  of  120°.  But  OQ,  OR  being  tangents,  the 
centres  of  the  two  spheres  (or  circular  arcs  in  the  figure)  lie  on 
perpendiculars  to  them ;   therefore  the  radii  CO,  CO  meet  at  an 


Fig.   108. 

angle  of  60°,  and  COC  is  an  equilateral  triangle.  That  is  to  say, 
the  centre  of  each  circle  hes  on  the  circumference  of  the  other; 
the  partition  lies  midway  between  the  two  centres ;  and  the 
length  (i.e.  the  diameter)  of  the  partition-wall,  PO,  is 

2  sin  60°  -  1-732 

times  the  radius,  or  -866  times  the  diameter,  of  each  of  the  cells. 
This  gives  us,  then,  the  form  of  an  aggregate  of  two  equal  cells 
under  uniform  conditions. 

As  soon  as  the  tensions  become  unequal,  whether  from  changes 
in  their  own  substance  or  from  differences  in  the  substances  with 
which  they  are  in  contact,  then  the  form  alters.     If  the  tension 

*  In  an  actual  calculation  we  must  of  course  always  take  account  of  the  tensions 
on  both  sides  of  each  film  or  membrane. 


302  THE   FORMS   OF   TISSUES  [ch. 

along  the  partition,  P,  diminishes,  the  partition  itself  enlarges, 
and  the  angle  QOR  increases :  until,  when  the  tension  P  is  very 
small  compared  to  Q  or  R,  the  whole  figure  becomes  a  circle,  and 
the  partition-wall,  dividing  it  into  two  hemispheres,  stands  at 
right  angles  to  the  outer  wall.  This  is  the  case  when  the  outer 
wall  of  the  cell  is  practically  solid.  On  the  other  hand,  if  P  begins 
to  increase  relatively  to  Q  and  R,  then  the  partition- wall  contracts, 
and  the  two  adjacent  cells  become  larger  and  larger  segments  of 
a  sphere,  until  at  length  the  system  becomes  divided  into  two 
separate  cells. 

In  the  spores  of  Liverworts  (such  as  Pellia),  the  first  partition- 
wall  (the  equatorial  partition  in  Fig.  109,  a)  divides  the  spore  into 
two  equal  halves,  and  is  therefore  a  plane  surface,  normal  to  the 
surface  of  the  cell ;    but  the  next  partitions  arise  near  to  either 


Fig.   109.     Spore  of  Pellia.     (After  Campbell.) 

end  of  the  original  spherical  or  elliptical  cell.  Each  of  these  latter 
partitions  \vill  (like  the  first)  tend  to  set  itself  normally  to  the 
cell-wall;  at  least  the  angles  on  either  side  of  the  partition  w^ll 
be  identical,  and  their  magnitude  will  depend  upon  the  tension 
existing  between  the  cell-wall  and  the  surrounding  medium. 
They  will  only  be  right  angles  if  the  cell- wall  is  already  practically 
solid,  and  in  all  probability  (rigidity  of  the  cell- wall  not  being 
quite  attained)  they  will  be  somewhat  greater.  In  either  case 
the  partition  itself  will  be  a  portion  of  a  sphere,  whose  curvature 
will  now  denote  a  difference  of  pressures  in  the  two  chambers  or 
cells,  which  it  serves  to  separate.  (The  later  stages  of  cell-division, 
represented  in  the  figures  h  and  c,  we  are  not  yet  in  a  position  to 
deal  with.) 

We  have  innumerable  cases,  near  the  tip  of  a  growing  filament, 
where  in  like  manner  the  partition- wall  which  cuts  ofE  the  terminal 


VIl] 


OF   CELL-PARTITIONS 


303 


cell  constitutes  a  spherical  lens-shaped  surface,  set  normally  to 
the  adjacent  walls.  At  the  tips  of  the  branches  of  many  Florideae, 
for  instance,  we  find  such  a  lenticular  partition.  In  Dictyota 
dichotoma,  as  figured  by  Reinke,  we  have  a  succession  of  such 
partitions ;  and,  by  the  way,  in  such  cases  as  these,  where  the 
tissues  are  very  transparent,  we  have  often  in  optical  section  a 
puzzling  confusion  of  lines ;  one  being  the  optical  section  of  the 
curved  partition- wall,  the  other  being  the  straight  linear  projection 
of  its  outer  edge  to  which  we  have  already  referred.  In  the 
conical  terminal  cell  of  Chara,  we  have  the  same  lens-shaped 
curve,  but  a  little  lower  down,  where  the  sides  of  the  shoot  are 
approximately  parallel,  we  have  flat  transverse  partitions,  at  the 
edges  of  which,  however,  we  recognise  a  convexity  of  the  outer 
cell-wall  and  a  definite  angle  of  contact,  equal  on  the  two  sides 
of  the  partition. 


Fig.   110.     Cells  of  Dictyota. 
(After  Reinke.) 


Fig.  HI. 


Terminal  and  other  cells 
of  Chara. 


In  the  young  antheridia  of  Chara  (Fig.  112),  and  in  the  not 
dissimilar  case  of  the  sporangium  (or  conidiophore)  of  Mucor,  we 
easily  recognise  the  hemispherical  form  of  the  septum  which  shuts 
off  the  large  spherical  cell  from  the  cylindrical 
filament.  Here,  in  the  first  phase  of  develop- 
ment, we  should  have  to  take  into  consideration 
the  different  pressures  exerted  by  the  single 
curvature  of  the  cylinder  and  the  double 
curvature  of  its  spherical  cap  (p.  221) ;  and 
we  should  find  that  the  partition  would  have 
a  somewhat  low  curvature,  with  a  radius  less 
than  the  diameter  of  the  cylinder;    which  it 

would    have    exactly    equalled    but    for    the       Fig-  112.    Young 

,  ,.   .         ,  .  ,  .    ,      .  .  antheridium  of 

adaitionai  pressure  inwards  which  it  receives  Chara. 


304 


THE   FORMS   OF  TISSUES 


[CH. 


from  the  curvature  of  the  large  surrounding  sphere.  But  as  the 
latter  continues  to  grow,  its  curvature  decreases,  and  so  likewise 
does  the  inward  pressure  of  its  surface ;  and  accordingly  the  little 
convex  partition  bulges  out  more  and  more. 

In  order  to  epitomise  the  foregoing  facts  let  the  annexed 
diagrams  (Fig.  113)  represent  a  system  of  three  films,  of  which 
one  is  a  partition- wall  between  the  other  two ;  and  let  the  tensions 
at  the  three  surfaces,  or  the  tractions  exercised  upon  a  point  at 
their  meeting-place,  be  proportional  to  T,  T'  and  t.  Let  a,  j3,  y 
be,  as  in  the  figure,  the  opposite  angles.     Then : 

(1)  If  T  be  equal  to  T',  and  t  be  relatively  insignificant, 
the  angles  a,  j8  will  be  of  90°. 


Fig.   113. 

(2)  If  T  =  T',  but  be  a  little  greater  than  t,  then  t  will  exert 
an  appreciable  traction,  and  a,  j8  will  be  more  than  90°,  say,  for 
instance,  100°. 

(3)  If  T- T' =  Athena,  ,^,y  will  all  equal  120°. 

The  more  complicated  cases,  when  t,  T  and  T'  are  all  unequal, 
are  already  sufficiently  explained. 

The  biological  facts  which  the  foregoing  considerations  go  a 
long  way  to  explain  and  account  for  have  been  the  subject  of  much 
argument  and  discussion,  especially  on  the  part  of  the  botanists. 
Let  me  recapitulate,  in  a  very  few  words,  the  history  of  this  long 
discussion. 

Some  fifty  years  ago,  Hofmeister  laid  it  down  as  a  general  law 
that  "  The  partition- wall  stands  always  perpendicular  to  what  was 
previously  the  principal  direction  of  growth  in  the  cell," — or,  in 
other  words,  perpendicular  to  the  long  axis  of  the  cell*.     Ten 


*  Hofmeister,  Pringsheini's  Jahrb.  in,  p.  272,  1863;    Hdb.  d. 
1867,  p.  129. 


Bot.  I, 


VII]  OF   CELL-PARTITIONS  305 

years  later,  Sachs  formulated  his  rule,  or  principle,  of  "  rectangular 
section,"  declaring  that  in  all  tissues,  however  complex,  the 
cell-walls  cut  one  another  (at  the  time  of  their  formation)  at  right 
andes*.  Years  before,  Schwendener  had  found,  in  the  final 
results  of  cell-division,  a  universal  system  of  "orthogonal  tra- 
jectoriest";  and  this  idea  Sachs  further  developed,  introducing 
complicated  systems  of  confocal  ellipses  and  hyperbolae,  and 
distinguishing  between  periclinal  walls,  whose  curves  approximate 
to  the  peripheral  contours,  radial  partitions,  which  cut  these  at 
an  angle  of  90°,  and  finally  antichnes,  which  stand  at  right  angles 
bo  the  other  two. 

Reinke,  in  1880,  was  the  first  to  throw  some  doubt  upon  this 
explanation.  He  pointed  out  various  cases  where  the  angle  was 
not  a  right  angle,  but  was  very  definitely  an  acute  one;  and 
he  saw,  apparently,  in  the  more  common  rectangular  symmetry 
merely  what  he  calls  a  necessary,  but  secondary,  result  of  growth  J. 

Within  the  next  few  years,  a  number  of  botanical  writers  were 
content  to  point  out  further  exceptions  to  Sachs's  Ilule§;  and  in 
some  cases  to  show  that  the  curvatures  of  the  partition- walls, 
especially  such  cases  of  lenticular  curvature  as  we  have  described, 
were  by  no  means  accounted  for  by  either  Hofmeister  or  Sachs ; 
while  within  the  same  period,  Sachs  himself,  and  also  Rauber, 
attempted  to  extend  the  main  generalisation  to  animal  tissues ||. 

While  these  writers  regarded  the  form  and  arrangement  of  the 
cell-walls  as  a  biological  phenomenon,  with  little  if  any  direct 
relation  to  ordinary  physical  laws,  or  with  but  a  vague  reference 
to  "  mechanical  conditions,"  the  physical  side  of  the  case  was 
soon  urged  by  others,  with  more  or  less  force  and  cogency.  Indeed 
the  general  resemblance  between  a  cellular  tissue  and  a  "froth"' 

*  Sachs,  Ueber  die  Anordnung  der  Zellen  in  jiingsten  Pflanzentheilen,  Verh. 
pkys.  med.  Ges.  Wurzburg,  xi,  pp.  219-242,  1877 ;  Uteber  Zellenanordnung  und 
Wachsthiim,  ibid,  xii,  1878;  Ueber  die  durch  Wachsthum  bedingte  Verschiebung 
kleinster  Theilchen  in  trajectorischen  Curven,  Monatsber.  k.  Akad.  Wiss.  Berlin, 
1880;  Physiology  of  Plants,  chap,  xxvii,  pp.  431-459,  Oxford,  1887. 

t  Schwendener,  Ueber  den  Bau  und  das  Wachsthum  des  Flechtenthallus, 
Naturf.  Ges.  Zurich,  Febr.   1860,  pp.  272-296. 

f  Reinke,  Lehrbuch  der  Botanik,  1880,  p.  519. 

§  Cf.  Leitgeb,  Uriters.  ilber  die  Lebermoose,  ii,  p.  4,  Graz,  1881. 

I!  Rauber,  Neue  Grundlegungen  zur  Kenntniss  der  Zelle,  Mar  ph.  Jahrb.  vnr, 
pp.  279,  334,  1882. 

T.  G.  20 


306  THE  FORMS   OF  TISSUES  [ch. 

had  been  pointed  out  long  before,  by  Melsens,  who  had  made  an 
"artificial  tissue"  by  blowing  into  a  solution  of  white  of  egg*. 

In  1886,  Berthold  pubUshed  his  Protoplasmamechanik,  in  which 
he  definitely  adopted  the  principle  of  "minimal  areas,"  and, 
following  on  the  lines  of  Plateau,  compared  the  forms  of  many 
cell-surfaces  and  the  arrangement  of  their  partitions  with  those 
assumed  under  surface  tension  by  a  system  of  "  weightless  films." 
But,  as  Klebs|  points  out  in  reviewing  Berthold's  book,  Berthold 
was  careful  to  stop  short  of  attributing  the  biological  phenomena 
to  a  definite  mechanical  cause.  They  remained  for  him,  as  they 
had  done  for  Sachs,  so  many  "phenomena  of  growth,"  or 
"properties  of  protoplasm." 

In  the  same  year,  but  while  still  apparently  unacquainted  with 
Berthold's  work,  ErreraJ  published  a  short  but  very  lucid  article, 
in  which  he  definitely  ascribed  to  the  cell-wall  (as  Hofmeister  had 
already  done)  the  properties  of  a  semi-liquid  film  and  drew  from 
this  as  a  logical  consequence  the  deduction  that  it  must  assume  the 
various  configurations  which  the  law  of  minimal  areas  imposes  on 
the  soap-bubble.  So  what  we  may  call  Errerd's  Law  is  formulated 
as  follows :  A  cellular  membrane,  at  the  moment  of  its  formation, 
tends  to  assume  the  form  which  would  be  assumed,  under  the 
same  conditions,  by  a  liquid  film  destitute  of  weight. 

Soon  afterwards  Chabry.  in  discussing  the  embryology  of  the 
Ascidians,  indicated  many  of  the  points  in  which  the  contacts 
between  cells  repeat  the  surface-tension  phenomena  of  the  soap- 
bubble,  and  came  to  the  conclusion  that  part,  at  least,  of  the 
embryological  phenomena  were  purely  physical  §;  and  the  same 
line  of  investigation  and  thought  were  pursued  and  developed  by 
Robert,  in  connection  with  the  embryology  of  the  Mollusca||. 
Driesch  again,  in  a  series  of  papers,  continued  to  draw  attention 
to  the  presence  of  capillary  phenomena  in  the  segmenting  cells 

*  C.  R.  Acad.  8c.  xxxiii,  p.  247,  1851;  Ann.  de  chimie  et  de  phys.  (3),  xxxm, 
p.  170,  1851;  Bull.  R.  Acad.  Belg.  xxiv,  p.  531,  1857. 

t  Klebs,  Biolog.  Centralbl.  vn,  pp.  193-201,  1^87. 

{  L.  Errera,  Sur  une  condition  fondamentale  d'equUibre  des  cellules  vivantes, 
C.  R.,  cni,  p.  822,  1886;  Bull.  Soc.  Beige  de  Microscopie,  xm,  Oct.  1886;  Recueil 
fTcBUvres  (Physiologie  generale),  1910,  pp.  201-205. 

§  L.  Chabry,  Embryologie  des  Ascidiens,  J.  Anat.  et  Physiol,  xxm,  p.  266,  1887. 

II  Robert,  Embryologie  des  Troques,  Arch,  de  Zool.  exp.  et  gen.  (3),  X,  1892. 


VIl] 


OF  EEKERA'S   LAW 


307 


of  various  embryos,  and  came  to  the  conclusion  that  the  mode  of 
segmentation  was  of  httle  importance  as  regards  the  final  result*. 
Lastly  de  Wildemanf,  in  a  somewhat  wider,  but  also  vaguer 
generalisation  than  Errera's,  declared  that  "The  form  of  the 
celhilar  framework  of  vegetables,  and  also  of  animals,  in  its 
essential  features,  depends  upon  the  forces  of  molecular  physics." 


Let  us  return  to  our  problem  of  the  arrangement  of  partition 
films.  When  we  have  three  bubbles  in  contact,  instead  of  two  as 
in  the  case  already  considered,  the  phenomenon  is  strictly  analogous 
to  our  former  case.  The  three  bubbles  will  be  separated  by  three 
partition  surfaces,  whose  curvature  will  depend  upon  the  relative 


Fisf.  Hi. 


size  of  the  spheres,  and  which  will  be  plane  if  the  latter  are  all  of 
the  same  dimensions ;  but  whether  plane  or  curved,  the  three 
partitions  will  meet  one  another  at  an  angle  of  120°,  in  an  axial 
line.  Various  pretty  geometrical  corollaries  accompany  this  ar- 
rangement. For  instance,  if  Fig.  114  represent  the  three  associated 
bubbles  in  a  plane  drawn  through  their  centres,  c,  c' ,  c"  (or  what 
is  the  same  thing,  if  it  represent  the  base  of  three  bubbles  resting 
on  a  plane),  then  the  Unes  uc,  uc" ,  or  sc,  sc' ,  etc.,  drawn  to  the 

*  "Dass  der  Furchungsmodus  etwas  fiir  das  Zukiinftige  unwesentliches  ist," 
Z.  f.  w.  Z.  Lv,  1893,  p.  37.  With  this  statement  compare,  or  contrast,  that  of 
Conkhn,  quoted  on  p.  4;    of.  also  pp.   157,  348  (footnotes). 

t  de  Wildeman,  Etudes  sur  I'attache  des  cloisons  cellulaires,  Mem.  Couronn. 
de  VAcad.  R.  de  Belgique,  liii,  84  pp.,  1893-4. 

20—2 


308 


THE   FORMS   OF  TISSUES 


[CH. 


centres  from  the  points  of  intersection  of  the  circular  arcs,  will 
always  enclose  an  angle  of  60°.  Again  (Fig.  115),  if  we  make  the 
angle  c"uf  equal  to  60°,  and  produce  uf  to  meet  cc"  in  /,/  will  be 
the  centre  of  the  circular  arc  which  constitutes  the  partition  Ou ; 
and  further,  the  three  points/,^,  h,  successively  determined  in  this 


I 


Fig.   115. 


manner,  will  lie  on  one  and  the  same  straight  line.  In  the  case 
of  coequal  bubbles  or  cells  (as  in  Fig.  114,  B),  it  is  obvious  that 
the  lines  joining  their  centres  form  an  equilateral  triangle;  and 
consequently,  that  the  centre  of  each  circle  (or  sphere)  lies  on  the 
circumference  of  the  other  two ;   it  is  also  obvious  that  uf  is  now 


VIl] 


OF   LAMARLE'S  LAW 


309 


parallel  to  cc" ,  and  accordingly  that  the  centre  of  curvature  of 
the  partition  is  now  infinitely  distant,  or  (as  we  have  already  said), 
that  the  partition  itself  is  plane. 

When  we  have  four  bubbles  in  conjunction,  they  would  seem 
to  be  capable  of  arrangement  in  two  symmetrical  ways :  either, 
as  in  Fig.  116  (A),  with  the  four  partition- walls  meeting  at  right 
angles,  or,  as  in  (B),  with^ve  partitions  meeting,  three  and  three, 
at  angles  of  120°.  This  latter  arrangement  is  strictly  analogous 
to  the  arrangement  of  three  bubbles  in  Fig.  114.  Now,  though 
both  of  these  figures,  from  their  symmetry,  are  apparently  figures  of 
equilibrium,  yet,  physically,  the  former  turns  out  to  be  of  unstable 


Fig.  116. 

and  the  latter  of  stable  equilibrium.  If  we  try  to  bring  our  four 
bubbles  into  the  form  of  Fig.  116,  A,  such  an  arrangement  endures 
only  for  an  instant ;  the  partitions  glide  upon  each  other,  a  median 
wall  springs  into  existence,  and  the  system  at  once  assumes  the 
form  of  our  second  figure  (B).  This  is  a  direct  consequence  of  the 
law  of  minimal  areas :  for  it  can  be  shewn,  by  somewhat  difficult 
mathematics  (as  was  first  done  by  Lamarle),  tbat,  in  dividing  a 
closed  space  into  a  given  number  of  chambers  by  means  of  partition- 
walls,  the  least  possible  area  of  these  partition- walls,  taken  together, 
can  only  be  attained  when  they  meet  together  in  groups  of  three, 
at  equal  angles,  that  is  to  say  at  angles  of  120°. 


310  THE  FORMS   OF  TISSUES  [ch. 

Wherever  we  have  a  true  cellular  complex,  an  arrangement  of 
cells  in  actual  physical  contact  by  means  of  a  boundary  film,  we 
find  this  general  principle  in  force;  we  must  only  bear  in  mind 
that,  for  its  perfect  recognition,  we  must  be  able  to  view  the 
object  in  a  plane  at  right  angles  to  the  boundary  walls.  For 
instance,  in  any  ordinary  section  of  a  vegetable  parenchyma,  we 
recognise  the  appearance  of  a  "froth,"  precisely  resembling  that 
which  we  can  construct  by  imprisoning  a  mass  of  soap-bubbles  in 
a  narrow  vessel  with  flat  sides  of  glass ;  in  both  cases  we  see  the 
cell-walls  everywhere  meeting,  by  threes,  at  angles  of  120°,  irre- 
spective of  the  size  of  the  individual  cells :  whose  relative  size,  on 
the  other  hand,  determines  the  curvature  of  the  partition- walls. 
On  the  surface  of  a  honey-comb  we  have  precisely  the  same 
conjunction,  between  cell  and  cell,  of  three  boundary  walls, 
meeting  at  120°.  In  embryology,  when  we  examine  a  segmenting 
egg,  of  four  (or  more)  segments,  we  find  in  like  manner,  in  the  great 
majority  of  cases,  if  not  in  all,  that  the  same  principle  is  still 
exemplified ;  the  four  segments  do  not  meet  in  a  common  centre, 
but  each  cell  is  in  contact  with  two  others,  and  the  three,  and  only 
three,  common  boundary  walls  meet  at  the  normal  angle  of  120°. 
A  so-called  polar  furrow*,  the  visible  edge  of  a  vertical  partition- 
wall,  joins  (or  separates)  the  two  triple  contacts,  precisely  as  in 
Fig.  116,  B. 

In  the  four-celled  stage  of  the  frog's  egg,  Rauber  (an  exception- 
ally careful  observer)  shews  us  three  alternative  modes  in  which 
the  four  cells  may  be  found  to  be  conjoined  (Fig.  117).  In  (A)  we 
have  the  commonest  arrangement,  which  is  that  which  we  have 
just  studied  and  found  to  be  the  simplest  theoretical  one;  that 
namely  where  a  straight  "polar  furrow"  intervenes,  and  where, 
at  its  extremities,  the  partition-walls  are  conjoined  three  by  three. 
In  (B),  we  have  again  a  polar  furrow,  which  is  now  seen  to  be  a 
portion  of  the  first  "segmentation-furrow"  (of.  Fig.  155  etc.)  by 
which  the  egg  was  originally  divided  into  two ;  the  four-celled 
stage  being  reached  by  the  appearance  of  the  transverse  furrows 

*  It  was  so  termed  by  Conklin  in  1897,  in  his  paper  on  Crepidula  (J.  of  Morph. 
xm,  1897).  It  is  the  Querfurche  of  Rabl  {Morph.  Jahrb.  v,  1879);  the  Polarfurche 
of  0.  Hertwig  {Jen.  Zeitschr.  xrv,  1880);  the  Brechungslinie  of  Rauber  (Neue 
Grundlage  zur  K.  der  ZeUe,  M.  Jb.  vin,  1882).  It  is  carefully  discussed  by  Robert, 
Dev.  des  Troques,  Arch,  de  Zool.  Exp.  et  Gen.  (3),  x,  1892,  p.  307  seq. 


VII]  OF   THE   POLAR  FURROW  311 

and  their  corresponding  partitions.  In  this  case,  the  polar 
furrow  is  seen  to  be  sinuously  curved,  and  Rauber  tells  us  that 
its  curvature  gradually  alters :  as  a  matter  of  fact,  it  (or  rather 
the  partition-wall  corresponding  to  it)  is  gradually  setting  itself 
into  a  position  of  equilibrium,  that  is  to  say  of  equiangular  contact 
with  its  neighbours,  which  position  of  equihbrium  is  already 
attained  or  nearly  so  in  Fig.  117,  A.  In  Fig.  117,  C,  we  have  a 
very  different  condition,  with  which  we  shall  deal  in  a  moment. 
According  to  the  relative  magnitude  of  the  bodies  in  contact, 
this  "polar  furrow"  may  be  longer  or  shorter,  and  it  may  be  so 
minute  as  to  be  not  easily  discernible ;  but  it  is  quite  certain  that 
no  simple  and  homogeneous  system  of  fluid  films  such  as  we 
are  deahng  with  is  in  equilibrium  without  its  presence.  In  the 
accounts  given,  however,  by  embryologists  of  the  segmentation  of 
the  egg.  while  the  polar  furrow  is  depicted  in  the  great  majority 


A  B  C 

Fig.   117.     Various  ways  in  which  the  four  cells  are  co-arranged  in 
the  four-celled  stage  of  the  frog's  egg.     (After  Rauber.) 

of  cases,  there  are  others  in  which  it  has  not  been  seen  and  some 
in  which  its  absence  is  definitely  asserted*.  The  cases  where  four 
cells,  lying  in  one  plane,  meet  in  a  point,  such  as  were  frequently 
figured  by  the  older  embryologists,  are  very  difficult  to  verify, 
and  I  have  not  come  across  a  single  clear  case  in  recent  literature. 
Considering  the  physical  stability  of  the  other  arrangement,  the 
great  preponderance  of  cases  in  which  it  is  known  to  occur,  the 
difficulty  of  recognising  the  polar  furrow  in  cases  where  it  is 
very  small  and  unless  it  be  specially  looked  for,  and  the  natural 
tendency  of  the  draughtsman  to  make  an  all  but  symmetrical 
structure  appear  wholly  so,  I  am  much  inclined  to  attribute  to 

*  Thus  Wilson  (J.  of  Morph.  vni,  1895)  declared  that  in  Amphioxus  the  polar 
furrow  was  occasionally  absent,  and  Driesch  took  occasion  to  criticise  and  to  throw 
doubt  upon  the  statement  {ArcJi.  f.  Entw.  Mech.  i,  1895,  j).  418). 


312  THE   FORMS   OF   TISSUES  [ch. 

error  or  imperfect  observation  all  those  cases  where  the  junction- 
lines  of  four  cells  are  represented  (after  the  manner  of  Fig.  116,  A) 
as  a  simple  cross* . 

But  while  a  true  four-rayed  intersection,  or  simple  cross,  is 
theoretically  impossible  (save  as  a  transitory  and  highly  unstable 
condition),  there  is  another  condition  which  may  closely  simulate 
it,  and  which  is  common  enough.  There  are  plenty  of  repre- 
sentations of  segmenting  eggs,  in  which,  instead  of  the  triple 
junction  and  polar  furrow,  the  four  cells  (and  in  Hke  manner  their 
more  numerous  successors)  are  represented  as  rounded  off,  and 
separated  from  one  another  by  an  empty  space,  or  by  a  httle  drop 
of  an  extraneous  fluid,  evidently  not  directly  miscible  with  the 
fluid  surfaces  of  the  cells.  Such  is  the  case  in  the  obviously 
accurate  figure  which  Ilauber  gives  (Fig.  117,  C)  of  the  third  mode 
of  conjunction  in  the  four-celled  stage  of  the  frog's  egg.  Here 
Rauber  is  most  careful  to  point  out  that  the  furrows  do  not  simply 
"cross,"  or  meet  in  a  point,  but  are  separated  by  a  little  space, 
which  he  calls  the  Polgrubchen,  and  asserts  to  be  constantly  present 
whensoever  the  polar  furrow,  or  Brechungslinie,  is  not  to  be 
discerned.  This  little  interposed  space,  with  its  contained  drop 
of  fluid,  materially  alters  the  case,  and  implies  a  new  condition 
of  theoretical  and  actual  equihbrium.  For,  on  the  one  hand,  we 
see  that  now  the  four  intercellular  partitions  do  not  meet  ove 
another  at  all ;  but  really  impinge  upon  four  new  and  separate 
partitions,  which  constitute  interfacial  contacts,  not  between  cell 
and  cell,  but  between  the  respective  cells  and  the  intercalated 
drop.  And  secondly,  the  angles  at  which  these  four  little  surfaces 
will  meet  the  four  cell-partitions,  will  be  determined,  in  the  usual 
way,  by  the  balance  between  the  respective  tensions  of  these  several 
surfaces.  In  an  extreme  case  (as  in  some  pollen-grains)  it  may  be 
found  that  the  cells  under  the  observed  circumstances  are  not  truly 
in  surface  contact :  that  they  are  so  many  drops  which  touch  but 
do  not  "wet"  one  another,  and  which  are  merely  held  together 
by  the  pressure  of  the  surrounding  envelope.     But  even  supposing, 

*  Precisely  the  same  remark  was  made  long  ago  by  Driesch:  "Das  so  oft 
schematisch  gezeichnete  VierzeUenstadium  mit  zwei  sich  in  zwei  Punkten  scheidende 
Medianen  kann  man  wohl  getrost  aus  der  Reihe  des  Existierenden  streichen," 
Entw.  mech.  Studien,  Z.  f.  w.  Z.  Lni,  p.  106,  1892.  Cf.  also  his  Math,  mechanische 
Bedeutung  morphologischer  Probleme  der  Biologie,  Jena,  59  pp.  1891. 


VII]  OF  THE   POLAE  FURROW  313 

as  is  in  all  probability  the  actual  case,  that  they  are  in  actual  fluid 
contact,  the  case  from  the  point  of  view  of  surface  tension  presents 
no  difficulty.  In  the  case  of  the  conjoined  soap-bubbles,  we  were 
dealing  with  similar  contacts  and  with  equal  surface  tensions  through- 
out the  system;  but  in  the  system  of  protoplasmic  cells  which 
constitute  the  segmenting  egg  we  must  make  allowance  for  an  in- 
equality of  tensions,  between  the  surfaces  where  cell  meets  cell,  and 
where  on  the  other  hand  cell-surface  is  in  contact  with  the  sur- 
rounding medium, — in  this  case  generally  water  or  one  of  the  fluids 
of  the  body.  Remember  that  our  general  condition  is  that,  in  our 
entire  system,  the  sum^  of  the  surface  energies  is  a  minimum ;  and, 
while  this  is  attained  by  the  sum 
of  the  surfaces  being  a  minimum 
in  the  case  where  the  energy  is 
uniformly  distributed,  it  is  not 
necessarily  so  under  non-uniform 
conditions.  In  the  diagram  (Fig. 
118)  if  the  energy  per  unit  area 
be  greater  along  the  contact 
surface  cc' ,  where  cell  meets  cell, 
than  along  ca  or  ch,  where  cell- 
surface  is  in  contact  with  the  surrounding  medium,  these  latter 
surfaces  will  tend  to  increase  and  the  surface  of  cell-contact 
to  diminish.  In  short  there  will  be  the  usual  balance  of  forces 
between  the  tension  along  the  surface  cc',  and  the  two  opposing 
tensions  along  ca  and  ch.  If  the  former  be  greater  than  either 
of  the  other  two,  the  outside  angle  will  be  less  than  120° ;  and  if 
the  tension  along  the  surface  cc'  be  as  much  or  more  than  the 
sum  of  the  other  two,  then  the  drops  will  stand  in  contact  only, 
save  for  the  possible  effect  of  external  pressure,  at  a  point.  This  is 
the  explanation,  in  general  terms,  of  the  peculiar  conditions 
obtaining  in  Nostoc  and  its  allies  (p.  300),  and  it  also  leads  us  to 
a  consideration  of  the  general  properties  and  characters  of  an 
"epidermal"  layer. 

While  the  inner  cells  of  the  honey-comb  are  symmetrically 
situated,  sharing  with  their  neighbours  in  equally  distributed 
pressures  or  tensions,  and  therefore  all  tending  with  great  accuracy 


314 


THE  JORMS   OF   TISSUES 


[CH. 


to  identity  of  form,  the  case  is  obviously  different  with  the  cells 
at  the  borders  of  the  system.  So  it  is,  in  like  manner,  with  our 
froth  of  soaprbubbles.  The  bubbles,  or  cells,  in  the  interior  of 
the  mass  are  all  alike  in  general  character,  and  if  they  be  equal 
in  size  are  alike  in  every  respect:  their  sides  are  uniformly 
flattened*,  and  tend  to  meet  at  equal  angles  of  120°.  But  the 
bubbles  which  constitute  the  outer  layer  retain  their  spherical 
surfaces,  which  however  still  tend  to  meet  the  partition-walls 
connected  with  them  at  constant  angles  of  120°.  This  outer  layer 
of  bubbles,  which  forms  the  surface  of  our  froth,  constitutes  after 
a  fashion  what  we  should  call  in  botany  an  "epidermal"  layer. 
But  in  our  froth  of  soap-bubbles  we  have,  as  a  rule,  the  same  kind 
of  contact  (that  is  to  say,  contact  with  air)  both  within  and  without 
the  bubbles ;  while  in  our  living  cell,  the  outer  wall  of  the  epidermal 
cell  is  exposed  to  air  on  the  one  side,  but  is  in  contact  with  the 


Fig.  119. 


protoplasm  of  the  cell  on  the  other :  and  this  involves  a  difference 
of  tensions,  so  that  the  outer  walls  and  their  adjacent  partitions 
are  no  longer  likely  to  meet  at  equal  angles  of  120°.  Moreover, 
a  chemical  change,  due  for  instance  to  oxidation  or  possibly  also 
to  adsorption,  is  very  likely  to  affect  the  external  wall,  and  may 
tend  to  its  consolidation ;  and  this  process,  as  we  have  seen,  is 
tantamount  to  a  large  increase,  and  at  the  same  time  an 
equalisation,  of  tension  in  that  outer  wall,  and  will  lead  the 
adjacent  partitions  to  impinge  upon  it  at  angles  more  and 
more  nearly  approximating  to  90° :  the  bubble-like,  or  spherical, 
surfaces  of  the  individual  cells  being  more  and  more  flattened 
in  consequence.  Lastly,  the  chemical  changes  which  affect  the 
outer  walls  of  the  superficial  cells  may  extend,  in  greater  or 
less  degree,  to  their  inner  walls  also :   with  the  result  that  these 

*  Compare,  however,  p.  299. 


VII]  OF  EPIDERMAL   TISSUES  315 

cells  will  tend  to  become  more  or  less  rectangular  throughout,  and 
will  cease  to  dovetail  into  the  interstices  of  the  next  subjacent 
layer.  These  then  are  the  general  characters  which  we  recognise 
in  an  epidermis ;  and  we  perceive  that  the  fundamental  character 
of  an  epidermis  simply  is  that  it  lies  on  the  outside,  and  that  its 
main  physical  characteristics  follow,  as  a  matter  of  course,  from 
the  position  which  it  occupies  and  from  the  various  consequences 
which  that  situation  entails.  We  have  however  by  no  means 
exhausted  the  subject  in  this  short  account;  for  the  botanist  is 
accustomed  to  draw  a  sharp  distinction  between  a  true  epidermis 
and  what  is  called  epidermal  tissue.  The  latter,  which  is  found  in 
such  a  sea-weed  as  Laminaria  and  in  very  many  other  cryptogamic 
plants,  consists,  as  in  the  hypothetical  case  we  have  described, 
of  a  more  or  less  simple  and  direct  modification  of  the  general  or 
fundamental  tissue.  But  a  "  true  epidermis,"  such  as  we  have  it 
in  the  higher  plants,  is  something  with  a  long  morphological  history, 
something  which  has  been  laid  down  or  differentiated  in  an  early 
stage  of  the  plant's  growth,  and  which  afterwards  retains  its 
separate  and  independent  character.  We  shall  see  presently  that 
a  physical  reason  is  again  at  hand  to  account,  under  certain 
circumstances,  for  the  early  partitioning  off,  from  a  mass  of 
embryonic  tissue,  of  an  outer  layer  of  cells  which  from  their  first 
appearance  are  marked  off  from  the  rest  by  their  rectangular  and 
flattened  form. 

We  have  hitherto  considered  our  cells,  or  bubbles,  as  lying  in 
a  plane  of  symmetry,  and  further,  we  have  only  considered  the 
appearance  which  they  present  as  projected  on  that  plane:  in 
simpler  words,  we  have  been  considering  their  appearance  in 
surface  or  in  sectional  view.  But  we  have  further  to  consider 
them  as  solids,  whether  they  be  still  grouped  in  relation  to  a  single 
plane  (like  the  four  cells  in  Fig.  116)  or  heaped  upon  one  another, 
as  for  instance  in  a  tetrahedral  form  like  four  cannon-balls ;  and  in 
either  case  we  have  to  pass  from  the  problems  of  plane  to  those  of 
solid  geometry.  In  short,  the  further  development  of  our  theme 
must  lead  us  along  two  paths  of  enquiry,  which  continually 
intercross,  namely  (1)  the  study  of  more  complex  cases  of  partition 
and  of  contact  in  a  plane,  and  (2)  the  whole  question  of  the  surfaces 


316  THE   FORMS   OF   TISSUES  [ch. 

and  angles  presented  by  solid  figures  in  symmetrical  juxtaposition. 
Let  us  take  a  simple  case  of  the  latter  kind,  and  again  afterwards, 
so  far  as  possible,  let  us  try  to  keep  the  two  themes  separate. 

Where  we  have  three  spheres  in  contact,  as  in  Fig.  114  or  in 
either  half  of  Fig.  116,  B,  let  us  consider  the  point  of  contact 
(0,  Fig.  114)  not  as  a  point  in  the  plane  section  of  the  diagram,  but 
as  a  point  where  three  furrows  meet  on  the  surface  of  the  system. 
At  this  point,  three  cells  meet ;  but  it  is  also  obvious  that  there  meet 
here  six  surfaces,  namely  the  outer,  spherical  walls  of  the  three 
bubbles,  and  the  three  partition-walls  which  divide  them,  two  and 
two.  Also,/oMr  lines  or  edges  meet  here ;  viz.  the  three  external  arcs 
which  form  the  outer  boundaries  of  the  partition- walls  (and  which 
correspond  to  what  we  commonly  call  the  "furrows"  in  the  seg- 
menting egg) ;  and  as  a  fourth  edge,  the  "arris"  or  junction  of  the 
three  partitions  (perpendicular  to  the  plane  of  the  paper),  where 
they  all  three  meet  together,  as  we  have  seen,  at  equal  angles  of 
120°.  Lastly,  there  meet  at  the  point  four  solid  angles,  each 
bounded  by  three  surfaces :  to  wit,  within  each  bubble  a  solid 
angle  bounded  by  two  partition- walls  and  by  the  surface  wall ; 
and  (fourthly)  an  external  solid  angle  bounded  by  the  outer 
surfaces  of  all  three  bubbles.  Now  in  the  case  of  the  soap-bubbles 
(whose  surfaces  are  all  in  contact  with  air,  both  outside  and  in), 
the  six  films  meeting  at  the  point,  whether  surface  films  or  partition 
films,  are  all  similar,  with  similar  tensions.  In  other  words  the 
tensions,  or  forces,  acting  at  the  point  are  all  similar  and  symmet- 
rically arranged,  and  it  at  once  follows  from  this  that  the  angles, 
solid  as  well  as  plane,  are  all  equal.  It  is  also  obvious  that,  as 
regards  the  point  of  contact,  the  system  will  still  be  symmetrical, 
and  its  symmetry  will  be  quite  unchanged,  if  we  add  a  fourth 
bubble  in  contact  with  the  other  three :  that  is  to  say,  if  where 
we  had  merely  the  outer  air  before,  we  now  replace  it  by  the  air 
in  the  interior  of  another  bubble.  The  only  difference  will  be  that 
the  pressure  exercised  by  the  walls  of  this  fourth  bubble  will  alter 
the  curvature  of  the  surfaces  of  the  others,  so  far  as  it  encloses 
them ;  and,  if  all  four  bubbles  be  identical  in  size,  these  surfaces 
which  formerly  we  called  external  and  which  have  now  come  to 
be  internal  partitions,  will,  like  the  others,  be  flattened  by  equal 
and  opposite  pressure,  into  planes.     We  are  now  dealing,  in  short. 


VII]  OF   TETRAHEDEAL   SYMMETRY  317 

with  six  planes,  meeting  symmetrically  in  a  point,  and  constituting 
there  four  equal  solid  angles. 

If  we  make  a  wire  cage,  in  the  form  of  a  regular  tetrahedron, 
and  dip  it  into  soap-solution,  then  when  we  withdraw  it  we  see 
that  to  each  one  of  the  six  edges  of  the  tetrahedron,  i.e.  to  each 
one  of  the  six  wires  which  constitute  the  little  cage,  a  film  has 
attached  itself ;  and  these  six  films  meet  internally  at  a  point,  and 
constitute  in  every  respect  the  symmetrical  figure  which  we  have 
just  been  describing.  In  short,  the  system  of  films  we  have 
hereby  automatically  produced  is  precisely  the  system  of  partition- 
walls  which  exist  in  our  tetrahedral  aggregation  of  four  spherical 


bubbles : — precisely  the  same,  that  is  to  say,  in  the  neighbourhood 
of  the  meeting-point,  and  only  differing  in  that  we  have  made  the 
wires  of  our  tetrahedron  straight,  instead  of  imitating  the  circular 
arcs  which  actually  form  the  intersections  of  our  bubbles.  This 
detail  we  can  easily  introduce  in  our  wire  model  if  we  please. 

Let  us  look  for  a  moment  at  the  geometry  of  our  figure.  Let  o 
(Fig.  120)  be  the  centre  of  the  tetrahedron,  i.e.  the  centre  of  sym- 
metry where  our  films  meet ;  and  let  oa,  oh,  oc,  od,  be  hues  drawn  to 
the  four  corners  of  the  tetrahedron.  Produce  ao  to  meet  the  base 
in  f,  then  a'pd  is  a  right-angled  triangle.  It  is  not  difficult  to 
prove  that  in  such  a  figure,  o  (the  centre  of  gravity  of  the  system) 


318  THE  FORMS   OF   TISSUES  [ch. 

lies  just  three-quarters  of  the  way  between  an  apex,  a,  and  a  point, 
J),  which  is  the  centre  of  gravity  of  the  opposite  base.     Therefore 

op  =  oa/3  =  od/3. 

Therefore  cos  dop  =  ^ ,  and  cos  aod  =  —  ^ . 

That  is  to  say,  the  angle  aod  is  just,  as  nearly  as  possible, 
109°  28'  16".  This  angle,  then,  of  109°  28'  16",  or  very  nearly 
109  degrees  and  a  half,  is  the  angle  at  which,  in  this  and  every 
other  solid  system  of  liquid  films,  the  edges  of  the  partition-walls 
meet  one  another  at  a  point.  It  is  the  fundamental  angle  in  the 
sohd  geometry  of  our  systems,  just  as  120°  was  the  fundamental 
angle  of  symmetry  so  long  as  we  considered  only  the  plane  pro- 
jection, or  plane  section,  of  three  films  meeting  in  an  edge. 

Out  of  these  two  angles,  we  may  construct  a  great  variety  of 
figures,  plane  and  solid,  which  become  all  the  more  varied  and 
complex  when,  by  considering  the  case  of  unequal  as  well  as  equal 
cells,  we  admit  curved  (e.g.  spherical)  as  well  as  plane  boundary 
surfaces.  Let  us  consider  some  examples  and  illustrations  of 
these,  beginning  with  those  which  we  need  only  consider  in  reference 
to  a  plane. 

Let  us  imagine  a  system  of  equal  cylinders,  or  equal  spheres, 
in  contact  with  one  another  in  a  plane,  and  represented  in  section 
by  the  equal  and  contiguous  circles  of  Fig.  121.  I  borrow  my 
figure,  by  the  way,  from  an  old  Italian  naturalist,  Bonanni  (a 
contemporary  of  BorelH,  of  Ray  and  Willoughby  and  of  Martin 
Lister),  who  dealt  with  this  matter  in  a  book  chiefly  devoted  to 
molluscan  shells*. 

It  is  obvious,  as  a  simple  geometrical  fact,  that  each  of  these 
equal  circles  is  in  contact  with  six  surrounding  circles.  Imagine 
now  that  the  whole  system  comes  under  some  uniform  stress. 
It  may  be  of  uniform  surface  tension  at  the  boundaries  of  all  the 
cells ;  it  may  be  of  pressure  caused  by  uniform  growth  or  expansion 
within  the  cells;  or  it  may  be  due  to  some  uniformly  applied 
constricting  pressure  from  without.  In  all  of  these  cases  the  points 
of  contact  between  the  circles  in  the  diagram  will  be  extended  into 

*  Ricreatione  delV  occhio  e  delta  mente,  nelV  Osservatione  delle  Chiocciole,  Roma, 
1681. 


VIl] 


OF   HEXAGONAL   SYMMETRY 


319 


lines  of  contact,  representing  surfaces  of  contact  in  the  actual 
spheres  or  cyUnders ;  and  the  equal  circles  of  our  diagram  will 
be  converted  into  regular  and  equal  hexagons.  The  angles  of 
these  hexagons,  at  each  of  which  three  hexagons  meet,  are  of 
course  angles  of  120°.  So  far  as  the  form  is  concerned,  so  long  as 
we  are  concerned  only  with  a  morphological  result  and  not  with 
a  physiological  process,  the  result  is  precisely  the  same  whatever 
be  the  force  which  brings  the  bodies  together  in  symmetrical 
apposition  ;  it  is  by  no  means  necessary  for  us,  in  the  first  instance, 
even   to   enquire   whether   it   be   surface   tension   or   mechanical 


Fig.   121.     Diagram  of  hexagonal  cells.     (After  Bonanni.) 

pressure  or  some  other  physical  force  which  is  the  cause,  or  the 
main  cause,  of  the  phenomenon. 

The  production  by  mutual  interaction  of  polyhedral  cells, 
which,  under  conditions  of  perfect  symmetry,  become  regular 
hexagons,  is  very  beautifully  illustrated  by  Prof.  Benard's 
"  tourhillons  cellulaires^^  (cf,  p.  259),  and  also  in  some  of  Leduc's 
diffusion  experiments.  A  weak  (5  per  cent.)  solution  of  gelatine 
is  allowed  to  set  on  a  plate  «f  glass,  and  little  drops  of  a  5  or 
10  per  cent,  solution  of  ferrocyanide  of  potassium  are  then  placed 
at  regular  intervals  upon  the  gelatine.  Immediately  each  little 
drop  becomes  the  centre,  or  pole,  of  a  system  of  diffusion  currents. 


320 


THE  FORMS   OF  TISSUES 


[CH. 


and  the  several  systems  conflict  with  and  repel  one  another,  so 
that  presently  each  little  area  becomes  the  seat  of  a  double  current 
system,  from  its  centre  outwards  and  back  again ;  until  at  length 
the  concentration  of  the  field  becomes  equalised  and  the  currents 


Fig.   122.     An  "artificial  tissue,"  formed  by  coloured  drops  of  sodium  chloride 
solution  diffusing  in  a  less  dense  solution  of  the  same  salt.     (After  Leduc.) 


Fig.  123.     An  artificial  cellular  tissue,  formed  by  the  diffusion  in  gelatine  of 
drops  of  a  solution  of  potassium  ferrocyanide.     (After  Leduc.) 


VII]  OF   HEXAGONAL   SYMMETRY  321 

cease.  After  equilibrium  is  attained,  and  when  the  gelatinous 
mass  is  permitted  to  dry,  we  have  an  artificial  tissue  of  more  or 
less  regularly  hexagonal  "cells,"  which  simulate  in  the  closest  way 
an  organic  parenchyma.  And  by  varying  the  experiment,  in  ways 
which  Leduc  describes,  we  may  simulate  various  forms  of  tissue, 
and  produce  cells  with  thick  walls  or  with  thin,  cells  in  close 
contact  or  with  wide  intercellular  spaces,  cells  with  plane  or  with 
curved  partitions,  and  so  forth. 

The  hexagonal  pattern  is  illustrated  among  organisms  in  count- 
less cases,  but  those  in  which  the  pattern  is  perfectly  regular,  by 
reason  of  perfect  uniformity  of  force  and  perfect  equality  of  the 
individual  cells,  are  not  so  numerous.  The  hexagonal  epithelium- 
cells  of  the  pigment  layer  of.  the  eye,  external  to  the  retina,  are 
a  good  example.  Here  we  have  a  single  layer  of  uniform  cells, 
reposing  on  the  one  hand  upon  a  basement  membrane,  supported 


Fig.   124.     Epidermis  of  Girardia.     (After  Goebel.) 

behind  by  the  solid  wall  of  the  sclerotic,  and  exposed  on  the  other 
hand  to  the  uniform  fluid  pressure  of  the  vitreous  humour.  The 
conditions  all  point,  and  lead,  to  a  perfectly  symmetrical  result : 
that  is  to  say,  the  cells,  uniform  in  size,  are  flattened  out  to  a 
uniform  thickness  by  the  fluid  pressure  acting  radially ;  and  their 
reaction  on  each  other  converts  the  flattened  discs  into  regular 
hexagons.  In  an  ordinary  columnar  epithelium,  such  as  that  of 
the  intestine,  we  see  again  that  the  columnar  cells  have  been 
compressed  into  hexagonal  prisms ;  but  here  as  a  rule  the  cells 
are  less  uniform  in  size,  small  cells  are  apt  to  be  intercalated 
among  the  larger,  and  the  perfect  symmetry  is  accordingly  lost. 
The  same  is  true  of  ordinary  vegetable  parenchyma ;  the  originally 
spherical  cells  are  approximately  equal  in  size,  but  only  approxi- 
mately ;  and  there  are  accordingly  all  degrees  in  the  regularity  and 
symmetry  of  the  resulting  tissue.  But  obviously,  wherever  we 
T.  G.  -  21 


322 


THE   FORMS   OF   TISSUES 


[CH. 


have,  in  addition  to  the  forces  which  tend  to  produce  the  regular 
hexagonal  symmetry,  some  other  asymmetrical  component  arising 
from  growth  or  traction,  then  our  regular  hexagons  will  be  dis- 
torted in  various  simple  ways.  This  condition  is  illustrated  in 
the  accompanying  diagram  of  the  epidermis  of  Girardia ;  it  also 
accounts  for  the  more  or  less  pointed  or  fusiform  cells,  each  still 
in  contact  (as  a  rule)  with  six  others,  which  form  the  epithelial 
lining  of  the  blood-vessels :  and  other  similar,  or  analogous, 
instances  are  very  common. 

In  a  soap-froth  imprisoned  between  two  glass  plates,  we  have 
a  symmetrical  system  of  cells,  which  appear  in  optical  section  (as 


Fig.   125.     Soap-froth  under  pressure.     (After  Rhumbler.) 

in  Fig.  125,  B)  as  regular  hexagons ;  but  if  we  press  the  plates  a 
little  closer  together,  the  hexagons  become  deformed  or  flattened 
(Fig.  125,  A).  In  this  case,  however,  if 
we  cease  to  apply  further  pressure,  the 
tension  of  the  films  throughout  the 
system  soon  adjusts  itself  again,  and  in  a 
short  time  the  system  has  regained  the 
former  symmetry  of  Fig.  125,  B. 

In  the  growth  of  an  ordinary  dicoty- 
ledonous leaf,  we  once  more  see  reflected  in 
the  form  of  its  epidermal  cells  the  tractions, 
irregular  but  on  the  whole  longitudinal, 
which  growth  has  superposed  on  the  ten- 
sions of  the  partition-walls  (Fig.  126).  In 
the  narrow  elongated  leaf  of  a  Monocoty- 
ledon, such  as  a  hyacinth,  the  elongated,  apparently  quadrangular 


Fig.  126.  From  leaf  of 
Elodea  canadensis.  (After 
Berthold.) 


VII]  OF   HEXAGONAL  SYMMETKY  323 

cells  of  the  epidermis  appear  as  a  necessary  consequence  of  the 
simpler  laws  of  growth  which  gave  its  simple  form  to  the  leaf  as 
a  whole.  In  this  last  case,  however,  as  in  all  the  others,  the  rule 
still  holds  that  only  three  partitions  (in  surface  view)  meet  in  a 
point;  and  at  their  point  of  meeting  the  walls  are  for  a  short 
distance  manifestly  curved,  so  as  to  permit  the  junction  to  take 
place  at  or  nearly  at  the  normal  angle  of  120°. 

Briefly  speaking,  wherever  we  have  a  system  of  cylinders  or 
spheres,  associated  together  with  sufficient  mutual  interaction  to 
bring  them  into  complete  surface  contact,  there,  in  section  or  in 
surface  view,  we  tend  to  get  a  pattern  of  hexagons. 

While  the  formation  of  an  hexagonal  pattern  on  the  basis  of  ready-formed 
and  symmetrically  arranged  material  units  is  a  very  common,  and  indeed  the 
general  way,  it  does  not  follow  that  there  are  not  others  by  which  such  a 
pattern  can  be  obtained.  For  instance,  if  we  take  a  little  triangular  dish  of 
mercury  and  set  it  vibrating  (either  by  help  of  a  tuning-fork,  or  by  simply 
tapping  on  the  sides)  we  shall  have  a  series  of  little  waves  or  ripples  starting 
inwards  from  each  of  the  three  faces ;  and  the  intercrossing,  or  interference 
of  these  three  sets  of  waves  procUices  crests  and  hollows,  and  intermediate 
points  of  no  disturbance,  whose  loci  are  seen  as  a  beautiful  pattern  of  minute 
hexagons.  It  is  possible  that  the  very  minute  and  astonishingly  regular 
pattern  of  hexagons  which  we  see,  for  instance,  on  the  surface  of  many  diatoms, 
may  be  a  phenomenon  of  this  order*.  The  same  maybe  the  case  also  inArcella, 
where  an  apparently  hexagonal  pattern  is  found  not  to  consist  of  simple 
hexagons,  but  of  "straight  lines  in  three  sets  of  parallels,  the  lines  of  each 
set  making  an  angle  of  sixty  degrees  with  those  of  the  other  two  sets  f."  We 
must  also  bear  in  mind,  in  the  case  of  the  minuter  forms,  the  large  possibilities 
of  optical  illusion.  For  instance,  in  one  of  Abbe's  "diffraction-plates,"  a 
pattern  of  dots,  set  at  equal  interspaces,  is  reproduced  on  a  very  minute  scale 
by  photography ;  but  under  certain  conditions  of  microscopic  illumination 
and  focussing,  these  isolated  dots  appear  as  a  pattern  of  hexagons. 

A  symmetrical  arrangement  of  hexagons,  such  as  we  have  just  been 
studying,  suggests  various  simple  geometrical  corollaries,  of  which  the  following 
may  perhaps  be  a  useful  one. 

We  may  sometimes  desire  to  estimate  the  number  of  hexagonal  areas  or 
facets  in  some  structure  where  these  are  numerous,  such  for  instance  as  the 

*  Cf.  some  of  J.  H.  Vincent's  photographs  of  ripples,  in  Phil.  Mag.  1897-1899; 
or  those  of  F.  R.  Watson,  in  Phys.  Review,  1897,  1901,  1916.  The  appearance  will 
depend  on  the  rate  of  the  wave,  and  in  turn  on  the  surface-tension;  with  a  low 
tension  one  would  probably  see  only  a  moving  "jabble."  FitzGerald  thought 
diatom-patterns  might  be  due  to  electromagnetic  vibrations  ( Wori.s,  p.  503,  1902). 

t  Cushman,  J.  A.  and  Henderson,  W.  P.,  Amer.  Nat.  xl,  pp.  797-802,  1906. 

21 2 


324  THE   FOErMS  OF  TISSUES  [ch 

cornea  of  an  insect's  eye,  or  in  the  minute  pattern  of  hexagons  on  many  diatoms. 
An  approximate  enumeration  is  easily  made  as  follows. 

For  the  area  of  a  hexagon  (if  we  call  8  the  short  diameter,  that  namely 
which  bisects  two  of  the  opposite  sides)  is  8^  x  ^3/2,  the  area  of  a  circle 
being  d'^ .  7r/4.  Then,  if  the  diameter  (d)  of  a  circular  area  include  n  hexagons, 
the  area  of  that  circle  equals  {n .  8)^  x  7r/4.  And,  dividing  this  number  by 
the  area  of  a  single  hexagon,  we  obtain  for  the  number  of  areas  in  the  circle, ' 
each  equal  to  a  hexagonal  facet,  the  expression  n^  x  7r/4  x  2/s^B  —  0-907»^,  or 
9/10 .  71^,  nearly. 

This  calculation  deals,  not  only  with  the  complete  facets,  but  with  the 
areas  of  the  broken  hexagons  at  the  periphery  of  the  circle.  If  we  neglect 
these  latter,  and  consider  our  whole  field  as  consisting  of  successive  rings  of 
hexagons  about  a  central  one,  we  may  obtain  a  still  simpler  rule*.  For 
obviously,  around  our  central  hexagon  there  stands  a  zone  of  six,  and  around 
these  a  zone  of  twelve,  and  around  these  a  zone  of  eighteen,  and  so  on.  And 
the  total  number,  excluding  the  central  hexagon,  is  accordingly: 


^'or  one  zone 

6 

=  2x3 

=  3x1x2, 

,,     two  zones 

18 

=  3x6 

=  3x2x3, 

,,     three  zones 

36 

=  4x9 

=  3x3x4, 

,,     four  zones 

60 

=  5  X  12 

=  3x4x5, 

,,     five  zones 

90 

=  6  X  15 

=  3x5x6, 

and  so  forth.  If  N  be  the  number  of  zones,  and  if  we  add  one  to  the  above 
numbers  for  the  odd  central  hexagon,  the  rule  evidently  is,  that  the  total 
number,  H,  =  3N  {N  +  1)  +  1.  Thus,  if  in  a  preparation  of  a  fly's  cornea, 
I  can  count  twenty-five  facets  in  a  line  from  a  central  one,  the  total  number 
in  the  entire  circular  field  is  (3  x  25  x  26)  +  1  =  1951t- 

The  same  principles  which  account  for  the  development  of 
hexagonal  symmetry  hold  true,  as  a  matter  of  course,  not  only 
of  individual  cells  (in  the  biological  sense),  but  of  any  close- 
packed  bodies  of  uniform  size  and  originally  circular  outline; 
and  the  hexagonal  pattern  is  therefore  of  very  common  occurrence, 
under  widely  different  circumstances.  The  curious  reader  may 
consult  Sir  Thomas  Browne's  quaint  and  beautiful  account,  in  the 
Garden  of  Cyrus,  of  hexagonal  (and  also  of  quincuncial)  symmetry 
in  plants  and  animals,  which  "doth  neatly  declare  how  nature 
Geometrizeth,  and  observeth  order  in  all  things." 

*  This  does  not  merely  neglect  the  broken  ones  but  all  whose  centres  He  between 
this  circle  and  a  hexagon  inscribed  in  it. 

f  For  more  detailed  calculations  see  a  paper  by  "H.M."  [?  H.  Munro],  in 
Q.  J.  M.  S.  VI,  p.  83,  1858. 


VIl] 


OF   HEXAGONAL  SYMMETRY 


325 


'We  have  many  varied  examples  of  this  principle  among  corals, 
wherever  the  polypes  are  in  close  juxtaposition,  with  neither 
empty  space  nor  accumulations  of  matrix  between  their  adjacent 
walls.  Favosites  gothlandica,  for  instance,  furnishes  us  with  an 
excellent  example.  In  the  great  genus  Lithostrotion  we  have  some 
species  that  are  "massive"  and  others  that  are  "fasciculate"  ;  in 
other  words  in  some  the  long  cylindrical  corallites  are  in  close  con- 
tact with  one  another,  and  in  others  they  are  separate  and  loosely 
bundled  (Fig.  127).     Accordingly  in  the  former  the  corallites  are 


Fig.   127.     Lithostrotion  Martini. 
(After  Nicholson.) 


Fig.   128.     Cyathophyllum  hexagonum. 
(From  Nicholson,  after  Zittel.) 


squeezed  into  hexagonal  prisms,  while  in  the  latter  they  retain  their 
cylindrical  form.  Where  the  polypes  are  comparatively  few,  and 
so  have  room  to  spread,  the  mutual  pressure  ceases  to  work  or 
only  tends  to  push  them  asunder,  letting  them  remain  circular  in 
outline  (e.g.  Thecosmiha).  Where  they  vary  gradually  in  size,  as 
for  instance  in  Cyathophyllum  hexagonum,  they  are  more  or  less 
hexagonal  but  are  not  regular  hexagons ;  and  where  there  is  greater 
and  more  irregular  variation  in  size,  the  cells  will  be  on  the 
average  hexagonal,  but  some  will  have  fewer  and  some  more  sides 
than  six,  as  in  the  annexed  figure  of  Arachnophyllum  (Fig.  129). 


326 


THE  FORMS   OF  TISSUES 


[CH. 


Where  larger  and  smaller  cells,  corresponding  to  two  different 
kinds  of  zooids,  are  mixed  together,  we  may  get  various  results. 
If  the  larger  cells  are  numerous  enough  to  be  more  or  less  in 
contact  with  one  another  (e.g.  various  Monticuliporae)  they  will 
be  irregular  hexagons,  while  the  smaller  cells  between  them  will 
be  crushed  into  all  manner  of  irregular  angular  forms.  If  on  the 
other  hand  the  large  cells  are  comparatively  few  and  are  large 
and  strong-walled  compared  with  their  smaller  neighbours,  then 
the  latter  alone  will  be  squeezed  into  hexagons,  while  the  larger 
ones  will  tend  to  retain  their  circular  outline  undisturbed  (e.g. 
Heliopora,  Heliolites,  etc.). 

When,  as  happens  in  certain  corals,,  the  peripheral  walls  or 


Fig.   129.     Amchnophylhnii  pentagoyium. 
(After  Nicholson.) 


"thecae"  of  the  individual  polypes  remain  undevelope'd  but 
the  radiating  septa  are  formed  and  calcified,  then  we  obtain  new 
and  beautiful  mathematical  configurations  (Fig.  131).  For  the 
radiating  septa  are  no  longer  confined  to  the  circular  or  hexagonal 
bounds  of  a  polypite,  but  tend  to  meet  and  become  confluent 
with  their  neighbours  on  every  side;  and,  tending  to  assume 
positions  of  equilibrium,  or  of  minimal  area,  under  the  restraints 
to  which  they  are  subject,  they  fall  into  congruent  curves;  and 
these  correspond,  in  a  striking  manner,  to  the  lines  of  force  running, 
in  a  common  field  of  force,  between  a  number  of  secondary  centres. 
Similar  patterns  may  be  produced  in  various  ways,  by  the  play 
of  osmotic  or  magnetic  forces ;  and  a  particular  and  very  curious 
case  is  to  be  found  in  those  complicated  forms  of  nuclear  division 


VII]  OF  HEXAGONAL  SYMMETRY  327 

known  as  triasters,  poly  asters,  etc.,  whose  relation  to  a  field  of 
force  Hartog  has  explained*.  It  is  obvious  that,  in  our  corals, 
these  curving  septa  are  all  orthogonal  to  the  non-existent  hexagonal 
boundaries.  As  the  phenomenon  is  wholly  due  to  the  imperfect 
development  or  non-existence  of  a  thecal  wall,  it  is  not  surprising 
that  we  find  identical  configurations  among  various  corals,  or 
families  of  corals,  not  otherwise  related  to  one  another;  we  find 
the  same  or  very  similar  patterns  displayed,  for  instance^  in 
Synhelia  (Oculinidae),  in  Phillipsastraea  {Rugosa),  in  Thamnas- 
traea  {Fungida),  and  in  many  more. 

The  most  famous  of  all  hexagonal  conformations  and  perhaps 
the  most  beautiful  is  that  of  the  bee's  cell.     Here  we  have,  as  in 


Fig.  131.     Surface-views  of  Corals  with  undeveloped  thecae  and  confluent  septa. 
A,  Thamnastraea ;    B,  Comoseris.     (From  Nicholson,  after  Zittel.) 

our  last  examples,  a  series  of  equal  cylinders,  compressed  by 
symmetrical  forces  into  regular  hexagonal  prisms.  But  in  this 
case  we  have  two  rows  of  such  cylinders,  set  opposite  to  one 
another,  end  to  end;  and  we  have  accordingly  to  consider  also 
the  conformation  of  their  ends.  We  may  suppose  our  original 
cylindrical  cells  to  have  spherical  ends,  which  is  their  normal  and 
symmetrical  mode  of  termination ;  and,  for  closest  packing,  it  is 
obvious  that  the  end  of  any  one  cylinder  will  touch,  and  fit  in 
between,  the  ends  of  three  cylinders  in  the  opposite  row.  It  is 
just  as  when  we  pile  round-shot  in  a  heap;    each  sphere  that  we 

*  Cf.  Hartog,  The  Dual  Force  of  the  Dividing  Cell,  Science  Progress  (n.s.),  i, 
Oct.  1907,  and  other  papers.  Also  Baltzer,  Ueber  mehrpolige  Mitosen  bei  Seeigel- 
eiern,  Inaug.  Diss.  1908. 


328  THEFOKMS   OF   TISSUES  [ch. 

set  down  fits  into  its  nest  between  three  others,  and  the  four 
form  a  regular  tetrahedral  arrangement.  Just  as  it  was  obvious, 
then,  that  by  mutual  pressure  from  the  six  laterally  adjacent  cells, 
any  one  cell  would  be  squeezed  into  a  hexagonal  prism,  so  is  it  also 
obvious  that,  by  mutual  pressure  against  the  three  terminal 
neighbours,  the  end  of  any  one  cell  will  be  compressed  into  a  solid 
trihedral  angle  whose  edges  will  meet,  as  in  the  analogous  case 
already  described  of  a  system  of  soap-bubbles,  at  a  plane  angle 
of  109°  and  so  many  minutes  and  seconds.  What  we  have  to 
comprehend,  then,  is  how  the  six  sides  of  the  cell  are  to  be  combined 
with  its  three  terminal  facets.  This  is  done  by  bevelling  off  three 
alternate  angles  of  the  prism,  in  a  uniform  manner,  until  we  have 
tapered  the  prism  to  a  point;  and  by  so  doing,  we  evidently 
produce  three  rhombic  surfaces,  each  of  which  is  double  of  the 
triangle  formed  by  joining  the  apex  to  the  three  untouched  angles 
of  the  prism.  If  we  experiment,  not  with  cylinders,  but  with 
spheres,  if  for  instance  we  pile  together  a  mass  of  bread-pills  (or 
pills  of  plasticine),  and  then  submit  the  whole  to  a  uniform  pressure, 
it  is  obvious  that  each  ball  (like  the  seeds  in  a  pomegranate,  as 
Kepler  said),  will  be  in  contact  with  twelve  others, — six  in  its  own 
plane,  three  below  and  three  above,  and  in  compression  it  will 
therefore  develop  twelve  plane  surfaces.  It  will  in  short  repeat, 
above  and  below,  the  conditions  to  which  the  bee's  cell  is  subject 
at  one  end  only;  and,  since  the  sphere  is  symmetrically  situated 
towards  its  neighbours  on  all  sides,  it  follows  that  the  twelve  plane 
sides  to  which  its  surface  has  been  reduced  will  be  all  similar, 
equal  and  similarly  situated.  Moreover,  since  we  have  produced 
this  result  by  squeezing  our  original  spheres  close  together,  it  is 
evident  that  the  bodies  so  formed  completely  fill  space.  The 
regular  solid  which  fulfils  all  these  conditions  is  the  rhombic 
dodecahedron.  The  bee's  cell,  then,  is  this  figure  incompletely 
formed :  it  is  a  hexagonal  prism  with  one  open  or  unfinished  end, 
and  one  trihedral  apex  of  a  rhombic  dodecahedron. 

The  geometrical  form  of  the  bee's  cell  must  have  attracted  the 
attention  and  excited  the  admiration  of  mathematicians  from  time 
immemorial.  Pappus  the  Alexandrine  has  left  us  (in  the  intro- 
duction to  the  Fifth  Book  of  his  Collections)  an  account  of  its 
hexagonal  plan,  and  he  drew  from  its  mathematical  symmetry  the 


VII]  OF   THE   BEE'S   CELL  329 

conclusion  that  the  bees  were  endowed  with  reason:  "There 
being,  then,  three  figures  which  of  themselves  can  fill  up  the 
space  round  a  point,  viz.  the  triangle,  the  square  and  the  hexagon, 
the  bees  have  wisely  selected  for  their  structure  that  which  contains 
most  angles,  suspecting  indeed  that  it  could  hold  more  honey  than 
either  of  the  other  two."  Erasmus  Barthohnus  was  apparently 
the  first  to  suggest  that  this  hypothesis  was  not  warranted,  and 
that  the  hexagonal  form  was  no  more  than  the  necessary  result 
of  equal  pressures,  each  bee  striving  to  make  its  own  little  circle 
as  large  as  possible. 

The  investigation  of  the  ends  of  the  cell  was  a  more  difficult 
matter,  and  came  later,  than  that  of  its  sides.  In  general  terms 
this  arrangement  was  doubtless  often  studied  and  described:  as 
for  instance,  in  the  Garden  of  Cyrus  '  "  And  the  Combes  them- 
selves so  regularly  contrived  that  their  mutual  intersections 
make  three  Lozenges  at  the  bottom  of  every  Cell ;  which  severally 
regarded  make  three  Rows  of  neat  Bhomboidall  Figures,  connected 
at  the  angles,  and  so  continue  three  several  chains  throughout  the 
whole  comb."  But  Maraldi*  (Cassini's  nephew)  was  the  first  to 
measure  the  terminal  solid  angle  or  determine  the  form  of  the 
rhombs  in  the  pyramidal  ending  of  the  cell.  He  tells  us  that  the 
angles  of  the  rhomb  are  110°  and  70°:  "Chaque  base  d'alveole 
est  formee  par  trois  rhombes  presque  toujours  egaux  et  semblables, 
qui,  suivant  les  mesures  que  nous  avons  prises,  ont  les  deux  angles 
obtus  chacun  de  110  degres,  et  par  consequent  les  deux  aigus 
chacun  de  70°."  He  also  stated  that  the  angles  of  the  trapeziums 
which  form  the  sides  of  the  body  of  the  cell  were  identical  angles, 
of  110°  and  70° ;  but  in  the  same  paper  he  speaks  of  the  angles  as 
being,  respectively,  109°  28'  and  70°  32'.  Here  a  singular  con- 
fusion at  once  arose,  and  has  been  perpetuated  in  the  books  f- 
"Unfortunately  Reaumur  chose  to  look  upon  this  second  deter- 
mination of  Maraldi's  as  being,  as  well  as  the  first,  a  direct  result 
of  measurement,  whereas  it  is  in  reality  theoretical.  He  speaks  of 
it  as  Maraldi's  more  precise  measurement,  and  this  error  has  been 
repeated  in  spite  of  its  absurdity  to  the  present  day ;    nobody 

*  Observations  sur  les  Abeilles,  Mem.  Acad.  Sc.  Paris,  1712,  p.  209. 

t  As  explained  by  Leslie  EUis,  in  his  essay  "On  the  Form  of  Bees'  Cells," 
in  Mathematical  and  other  Writings,  1853,  p.  353;  cf.  0.  Terquem,  Nouv.  Ann. 
Math.  1856,  p.  178. 


330 


THE   FORMS   OF   TISSUES 


[CH. 


appears  to  have  thought  of  the  impossibility  of  measuring  such  a 
thing  as  the  end  of  a  bee's  cell  to  the  nearest  minute."  At  any 
rate,  it  now  occurred  to  Reaumur  (as  curiously  enough,  it  had  not 
done  to  Maraldi)  that,  just  as  the  closely  packed  hexagons  gave 
the  minimal  extent  of  boundary  in  a  plane,  so  the  actual  solid 
figure,  as  determined  by  Maraldi,  might  be  that  which,  for  a  given 
solid  content,  gives  the  minimum  of  surface :  or  which,  in  other 
words,  would  hold  the  most  honey  for  the  least  wax.  He  set  this 
problem  before  Koenig,  and  the  geometer  confirmed  his  conjecture, 
the  result  of  his  calculations  agreeing  within  two  minutes  (109°  26' 
and  70°  34')  with  Maraldi's  determination.  But  again,  Maclaurin* 
and  LhuiUert,  by  different  methods,  obtained  a  result  identical 
with  Maraldi's ;  and  were  able  to  shew  that  the  discrepancy  of 
2'  was  due  to  an  error  in^  Koenig's  calculation  (of  tan  6  =  \/2), 
y  — that    is  to    say    to   the    imper- 

fection of  his  logarithmic  tables, — 
not  (as  the  books  say  J)  "to  a 
mistake  on  the  part  of  the  Bee." 
"Not  to  a  mistake  on  the  part  of 
Maraldi"  is,  of  course,  all  that  we 
are  entitled  to  say. 

The  theorem  may  be  proved  as 
follows : 

ABCDEF,  abcdef,  is  a  right 
prism  upon  a  regular  hexagonal  base. 
The  corners  BDF  are  cut  ofE  by 
planes  through  the  lines  AC,  CE, 
EA,  meeting  in  a  point  V  on  the 
axis  VN  of  the  prism,  and  intersect- 
ing Bh,  Dd,  Ff,  at  X,  Y,  Z.  It  is 
evident  that  the  volume  of  the  figure 
thus  formed  is  the  same  as  that  of 

the  original  prism  with  hexagonal 
Fig.  132.  &  r  & 

ends.     For,   if    the    axis    cut    the 

hexagon  ABCDEF  in  N,  the  volumes  ACVN,  ACBX  are  equal. 

*  Phil.  Trans,  xlu,  1743,  pp.  5(55-571.  t  ^^em.  de  VAcad.  de  Beilin,  1781. 

J  Cf.  Gregory,  Examples,  p.  106,  Wood's  Homes  without  Hands,  1865,  p.  428, 
Mach,  Science  of  Mechanics,  1902,  p.  453,  etc.,  etc. 


/r- 

\ 

0 

^\/\- 

"'■L. 

Y 

V 

\ 

\ 

F 

\         "^ 

c^ 

r 

\ 

VII]  OF   THE   BEE'S   CELL  331 

It  is  required  to  find  the  inclination  of  the  faces  forming  the 
trihedral  angle  at  V  to  the  axis,  such  that  the  surface  of  the 
figure  may  be  a  minimum. 

Let  the  angle  NVX,  which  is  half  the  solid  angle  of  the  prism, 
=  6 ;  the  side  of  the  hexagon,  as  AB,  =  a ;  and  the  height,  as 
Aa,  =  h. 

Then,  AC  =  2a  cos  30°  =  a^/3. 

And  VX  =  a/sin  6  (from  inspection  of  the  triangle  LXB) 

Therefore  the  area  of  the  rhombus  VAXC  =  a^\/3/2  sin  6. 

And  the  area  of  AabX  =  a/2  {2h  -  |FZ  cos  d) 
=  a/2  {2Ji-al2.cotd). 

Therefore  the  total  area  of  the  figure 

=  hexagon  abcdef  +  3a  [  2h  —  ^  cot  ^ )  +  3  „ . 

Therefore      ^  (^""^^^  -  ^""^  f  ^-^        a/S^os^N 
inereiore      — ^^       -    ^    l^sin^  ^      '  sin^  ^  / ' 

But  this  expression  vanishes,  that  is  to  say,  d  (Area)/(^^  =  0, 
when  cos  6  -  1/^3,  that  is  when  d  =  54°  44'  8" 

=  1  (109°  28'  16"). 

This  then  is  the  condition  under  which  the  total  area  of  the 
figure  has  its  minimal  value. 

That  the  beautiful  regularity  of  the  bee's  architecture  is  due 
to  some  automatic  play  of  the  physical  forces,  and  that  it  were 
fantastic  to  assume  (with  Pappus  and  Reaumur)  that  the  bee 
intentionally  seeks  for  a  method  of  economising  wax,  is  certain, 
but  the  precise  manner  of  this  automatic  action  is  not  so  clear. 
When  the  hive-bee  builds  a  solitary  cell,  or  a  small  cluster  of  cells, 
as  it  does  for  those  eggs  which  are  to  develop  into  queens,  it  makes 
but  a  rude  production.  The  queen-cells  are  lumps  of  coarse  wax 
hollowed  out  and  roughly  bitten  into  shape,  bearing  the  marks  of 
the  bee's  jaws,  like  the  marks  of  a  blunt  adze  on  a  rough-hewn  log. 
Omitting  the  simplest  of  all  cases,  when  (as  among  some  humble- 
bees)  the  old  cocoons  are  used  to  hold  honey,  the  cells  built  by 
the  "soHtary"  wasps  and  bees  are  of  various  kinds.  They  may 
be  formed  by  partitioning  off  httle  chambers  in  a  hollow  stem; 


332  THE   FORMS   OF  TISSUES  [ch. 

they  may  be  rounded  or  oval  capsules,  often  very  neatly  con- 
structed, out  of  mud,  or  vegetable^6re  or  little  stones,  agglutinated 
together  with  a  salivary  glue;  but  they  shew,  except  for  their 
rounded  or  tubular  form,  no  mathematical  symmetry.  The  social 
wasps  and  many  bees  build,  usually  out  of  vegetable  matter 
chewed  into  a  paste  with  saliva,  very  beautiful  nests  of  ''combs'  ; 
and  the  close-set  papery  cells  which  constitute  these  combs  are 
just  as  regularly  hexagonal  as  are  the  waxen  cells  of  the  hive-bee. 
But  in  these  cases  (or  nearly  all  of  them)  the  cells  are  in  a  single 
row ;  their  sides  are  regularly  hexagonal,  but  their  ends,  from  the 
want  of  opponent  forces,  remain  simply  spherical.  In  Melipona 
domestica  (of  which  Darwin  epitomises  Pierre  Huber's  description) 
"the  large  waxen  honey-cells  are  nearly  spherical,  nearly  equal  in 
size,  and  are  aggregated  into  an  irregular  mass."  But  the  spherical 
form  is  only  seen  on  the  outside  of  the  mass ;  for  inwardly  each 
cell  is  flattened  into  "two,  three  or  more  flat  surfaces,  according 
as  the  cell  adjoins  two,  three  or  more  other  cells.  When  one  cell 
rests  on  three  other  cells,  which  from  the  spheres  being  nearly 
of  the  same  size  is  very  frequently  and  necessarily  the  case,  the 
three  flat  surfaces  are  united  into  a  pyramid  ;  and  this  pyramid,  as 
Huber  has  remarked,  is  manifestly  a  gross  imitation  of  the  three- 
sided  pyramidal  base  of  the  cell  of  the  hive-bee*."  The  question 
is,  to  what  particular  force  are  we  to  ascribe  the  plane  surfaces 
and  definite  angles  which  define  the  sides  of  the  cell  in  all  these 
cases,  and  the  ends  of  the  cell  in  cases  where  one  row  meets  and 
opposes  another.  We  have  seen  that  BarthoKn  suggested,  and  it 
is  still  commonly  beheved,  that  this  result  is  due  to  simple  physical 
pressure,  each  bee  enlarging  as  much  as  it  can  the  cell  which  it 
is  a-building,  and  nudging  its  wall  outwards  till  it  fills  every 
intervening  gap  and  presses  hard  against  the  similar  efforts  of 
its  neighbour  in  the  cell  next  door|.     But  it  is  very  doubtful 

*  Origin  of  Species,  ch.  vni  (6th  ed.,  p.  221).  The  cells  of  various  bees, 
humble-bees  and  social  wasps  have  been  described  and  mathematically  investigated 
by  K.  Miillenhoijf,  Pfliiger's  Archiv  xxxii,  p.  589,  1883;  but  his  many  interesting 
results  are  too  complex  to  epitomise.  For  figures  of  various  nests  and  combs  see 
(e.g.)  von  Biittel-Reepen,  Biol.  Centralbl.  xxxiii,  pp.  4,  89,  129,  183,  1903. 

t  Darwin  had  a  somewhat  similar  idea,  though  he  allowed  more  play  to  the 
bee's  instinct  or  conscious  intention.  Thus,  when  he  noticed  certain  half-completed 
cell-walls  to  be  concave  on  one  side  and  convex  on  the  other,  but  to  become  perfectly 
flat  when  restored  for  a  short  time  to  the  hive,  he  says:    "It  was  absolutely  im- 


VII]  OF   THE   BEE'S   CELL  333 

whether  such  physical  or  mechanical  pressure,  more  or  less  inter- 
mittently exercised,  could  produce  the  all  but  perfectly  smooth, 
plane  surfaces  and  the  all  but  perfectly  definite  and  constant 
angles  which  characterise  the  cell,  whether  it  be  constructed  of 
wax  or  papery  pulp.  It  seems  more  likely  that  we  have  to  do 
with  a  true  surface-tension  efi'ect ;  in  other  words,  that  the  walls 
assume  their  configuration  when  in  a  semi-fluid  state,  while  the 
papery  pulp  is  still  liquid,  or  while  the  wax  is  warm  under  the 
high  temperature  of  the  crowded  hive*.  Under  these  circum- 
stances, the  direct  efforts  of  the  wasp  or  bee  may  be  supposed 
to  be  limited  to  the  making  of  a  tubular  cell,  as  thin  as  the  nature 
of  the  material  permits,  and  packing  these  little  cells  as  close  as 
possible  together.  It  is  then  easily  conceivable  that  the  sym- 
metrical tensions  of  the  adjacent  films  (though  somewhat  retarded 
by  viscosity)  should  suffice  to  bring  the  whole  system  into  equi- 
librium, that  is  to  say  into  the  precise  configuration  which  the 
comb  actually  presents.  In  short,  the  Maraldi  pyramids  which 
terminate  the  bee's  cell  are  precisely  identical  with  the  facets  of 
a  rhombic  dodecahedron,  such  as  we  have  assumed  to  constitute 
(and  which  doubtless  under  certain  conditions  do  constitute)  the 
surfaces  of  contact  in  the  interior  of  a  mass  of  soap-bubbles  or 
of  uniform  parenchymatous  cells ;  and  there  is  every  reason  to 
believe  that  the  physical  explanation  is  identical,  and  not  merely 
mathematically  analogous. 

The  remarkable  passage  in  which  Bufton  discusses  the  bee's 
cell  and  the  hexagonal  configuration  in  general  is  of  such  historical 
importance,  and  tallies  so  closely  with  the  whole  trend  of  our 
enquiry,  that  I  will  quote  it  in  full:  "Dirai-je  encore  un  mot; 
ces  cellules  des  abeilles,  tant  vantees,  tant  admirees,  me  fournissent 
une  preuve  de  plus  contre  I'enthousiasme  et  I'admiration ;  cette 
figure,  toute  geometrique  et  toute  reguliere  qu'elle  nous  parait,  et 
qu'elle  est  en  efEet  dans  la  speculation,  n'est  ici  qu'un  resultat 
mecanique  et  assez  imparfait  qui  se  trouve  souvent  dans  la  nature, 

possible,  from  the  extreme  thinness  of  the  Uttle  plate,  that  they  could  have  effected 
this  by  gnawing  away  the  convex  side;  and  I  suspect  that  the  bees  in  such  cases 
stand  on  opposite  sides  and  push  and  bend  the  ductile  and  warm  wax  (which  as 
I  have  tried  is  easily  done)  into  its  proper  intermediate  plane,  and  thus  flatten  it." 
*  Since  writing  the  above,  I  see  that  MiillenhofE  gives  the  same  explanation, 
and  declares  that  the  waxen  wall  is  actually  a  Flussigkeitshdutchen,  or  liquid  film. 


334  THE  FORMS   OF   TISSUES  [ch. 

et  que  Ton  remarque  meme  dans  les  productions  les  plus  brutes ; 
les  cristaux  et  plusieurs  autres  pierres,  quelques  sels,  etc.,  prennent 
constamment  cette  figure  dans  leur  formation.  Qu'on  observe  les 
petites  ecailles  de  la  peau  d'une  roussette,  on  verra  qu'elles  sont 
hexagones,  parce  que  chaque  ecaille  croissant  en  meme  temps  se 
fait  obstacle,  et  tend  a  occuper  le  plus  d'espace  qu'il  est  possible 
dans  un  espace  donne :  on  voit  ces  memes  hexagones  dans  le 
second  estomac  des  animaux  ruminans,  on  les  trouve  dans  les 
graines,  dans  leurs  capsules,  dans  certaines  fieurs,  etc.  Qu'on 
remplisse  un  vaisseau  de  pois,  ou  plutot  de  quelque  autre  graine 
cylindrique,  et  qu'on  le  ferme  exactement  apres  y  avoir  verse 
autant  d'eau  que  les  intervalles  qui  restent  entre  ces  graines 
peuvent  en  recevoir;  qu'on  fasse  bouillir  cette  eau,  tous  ces 
cylindres  deviendront  de  colonnes  a  six  pans*.  On  y  voit  claire- 
ment  la  raison,  qui  est  purement  mecanique ;  chaque  graine,  dont 
la  figure  est  cylindrique,  tend  par  son  renfiement  a  occuper  le 
plus  d'espace  possible  dans  un  espace  donne,  elles  deviennent  done 
toutes  necessairement  hexagones  par  la  compression  reciproque. 
Chaque  abeille  cherche  a  occuper  de  meme  le  plus  d'espace  possible 
dans  un  espace  donne,  il  est  done  necessaire  aussi,  puisque  le 
corps  des  abeilles  est  cylindrique,  que  leurs  cellules  sont  hexagones, 
— par  la  meme  raison  des  obstacles  reciproques.  On  donne  plus 
d'esprit  aux  mouches  dont  les  ouvrages  sont  les  plus  reguliers; 
les  abeilles  sont,  dit-on,  plus  ingenieuses  que  les  guepes,  que  les 
frelons,  etc.,  qui  savent  aussi  I'architecture,  mais  dont  les  con- 
structions sont  plus  grossieres  et  plus  irregulieres  que  celles  des 
abeilles :  on  ne  veut  pas  voir,  ou  Ton  ne  se  doute  pas  que  cette 
regularite,  plus  ou  moins  grande,  depend  uniquement  du  nombre 
et  de  la  figure,  et  nullement  de  I'intelligence  de  ces  petites  betes ; 
plus  elles  sont  nombreuses,  plus  il  y  a  des  forces  qui  agissent 
egalement  et  s'opposent  de  meme,  plus  il  y  a  par  consequent  de 
contrainte  mecanique,  de  regularite  forcee,  et  de  perfection 
apparente  dans  leurs  productions  f." 

*  Bonnet  criticised  Buffon's  explanation,  on  the  ground  that  his  description 
was  incomplete ;   for  Buffon  took  no  account  of  the  Maraldi  pyramids. 

t  Buffon,  Histoire  Naturelle,  rv,  p.  99.  Among  many  other  papers  on  the 
Bee's  cell,  see  Barclay,  Mem.  Wernerian  Soc.  ir,  p.  25!t  (1812),  1818;  Sharpe,  Phil. 
Mag.  IV,  1828,  pp.  19-21;  L.  Lalanne,  Ann.  Sci.  Nat.  (2)  Zool.  xm,  pp.  358-374, 
1840;   Haughton,  Ann.  Mag.  Nat  Hist.  (3),  xi,  pp.  415-429,  1863;   A.  R.  Wallace, 


VIl] 


OF   HEXAGONAL   SYMMETRY 


335 


A  very  beautiful  hexagonal  symmetry,  as  seen  in  section,  or 
dodecahedral,  as  viewed  in  the  solid,  is  presented  by  the  cells 
which  form  the  pith  of  certain  rushes  (e.g.  Juncus  ejfusus),  and 
somewhat  less  diagrammatically  by  those'  which  make  the  pith 
of  the  banana.  These  cells  are  stellate  in  form,  and  the  tissue 
presents  in  section  the  appearance  of  a  network  of  six-rayed 
stars  (Fig.  133,  c),  linked  together  by  the  tips  of  the  rays,  and 
separated  by  symmetrical,  air-filled,  intercellular  spaces.  In  thick 
sections,  the  solid  twelve-rayed  stars  may  be  very  beautifully  seen 
under  the  binocular  microscope. 


^^^M 

:  c 


Fig.  133.  Diagram  of  development  of  "stellate  cells,"  in  pith  of  Juncuf>. 
(The  dark,  or  shaded,  areas  represent  the  cells;  the  Ught  areas  being  the 
gradually  enlarging  "intercellular  spaces.") 


What  has  happened  here  is  not  difficult  to  understand. 
Imagine,  as  before,  a  system  of  equal  spheres  all  in  contact,  each 
one  therefore  touching  six  others  in  an  equatorial  plane ;  and  let 
the  cells  be  not  only  in  contact,  but  become  attached  at  the  points 
of  contact.  Then  instead  of  each  cell  expanding,  so  as  to  encroach 
on  and  fill  up  the  intercellular  spaces,  let  each  cell  tend  to  contract 
or  shrivel  up,  by  the  withdrawal  of  fluid  from  its  interior.     The 

ibid.  XII,  p.  303,  1863;    Jeffries  Wyman.  Pr.  Anier.  Acad,  of  Arts  and  8c.  vir,  pp. 
68-83,  1868;  Chauncey  Wright,  ibid,  iv,  p.  432,  1860. 


336  THE  FORMS   OF   TISSUES  [ch. 

result  will  obviously  be  that  the  intercellular  spaces  will  increase ; 
the  six  equatorial  attachments  of  each  cell  (Fig.  133,  a)  (or  its  twelve 
attachments  in  all,  to  adjacent  cells)  will  remain  fixed,  and  the 
portions  of  cell-wall  between  these  points  of  attachment  will  be 
withdrawn  in  a  symmetrical  fashion  (b)  towards  the  centre.  As 
the  final  result  (c)  we  shall  have  a  " dodecahedral  star"  or  star- 
polygon,  which  appears  in  section  as  a  six-rayed  figure.  It  is 
obviously  necessary  that  the  pith-cells  should  not  only  be  attached 
to  one  another,  but  that  the  outermost  layer  should  be  firmly 
attached  to  a  boundary  wall,  so  as  to  preserve  the  symmetry  of 
the  system.  What  actually  occurs  in  the  rush  is  tantamount  to 
this,  but  not  absolutely  identical.  Here  it  is  not  so  much  the 
pith-cells  which  tend  to  shrivel  within  a  boundary  of  constant 
size,  but  rather  the  boundary  wall  (that  is,  the  peripheral  ring  of 
woody  and  other  tissues)  which  continues  to  expand  after  the 
pith-cells  which  it  encloses  have  ceased  to  grow  or  to  multiply. 
The  twelve  points  of  attachment  on  the  spherical  surface  of  each 
little  pith-cell  are  uniformly  drawn  asunder ;  but  the  content,  or 
volume,  of  the  cell  does  not  increase  correspondingly;  and  the 
remaining  portions  of  the  surface,  accordingly,  shrink  inwards  and 
gradually  constitute  the  complicated  surface  of  a  twelve-pointed 
star,  which  is  still  a  symmetrical  figure  and  is  still  also  a  surface 
of  minimal  area  under  the  new  conditions. 

A  few  years  after  the  publication  of  Plateau's  book,  Lord 
Kelvin  shewed,  in  a  short  but  very  beautiful  paper  *,  that  we  must 
not  hastily  assume  from  such  arguments  as  the  foregoing,  that 
a  close-packed  assemblage  of  rhombic  dodecahedra  will  be  the  true 
and  general  solution  of  the  problem  of  dividing  space  with  a 
minimum  partitional  area,  or  will  be  present  in  a  cellular  liquid 
"foam,"  in  which  it  is  manifest  that  the  problem  is  actually  and 
automatically  solved.  The  general  mathematical  solution  of  the 
problem  (as  we  have  already  indicated)  is,  that  every  interface  or 
partition- wall  must  have  constant  curvature  throughout;  that 
where  such  partitions  meet  in  an  edge,  they  must  intersect  at 
angles  such  that  equal  forces,  in  planes  perpendicular  to  the  line 

*  Sir  W.  Thomson,  On  the  Division  of  Space  with  Minimum  Partitional  Area, 
Phil.  Mag.  (5),  xxrv,  pp.  503-.514,  Dec.  1887;   cf.  Baltimore  Lectures,  1904,  p.  615. 


VIl] 


OF   THE   PARTITIONING   OF  SPACE 


337 


of  intersection,  shall  balance ;  and  finally,  that  no  more  than  three 
such  interfaces  may  meet  in  a  line  or  edge,  whence  it  follows  that 
the  angle  of  intersection  of  the  film-surfaces  must  be  exactly  120°. 
An  assemblage  of  equal  and  similar  rhombic  dodecahedra  goes  far 
to  meet  the  case :  it  completely  fills  up  space ;  all  its  surfaces  or 
interfaces  are  planes,  that  is  to  say,  surfaces  of  constant  curvature 
throughout ;  and  these  surfaces  all  meet  together  at  angles  of  120°. 
Nevertheless,  the  proof  that  our  rhombic  dodecahedron  (such  as 
we  find  exemplified  in  the  bee's  cell)  is  a  surface  of  minimal  area, 
is  not  a  comprehensive  proof;  it  is  limited  to  certain  conditions, 
and  practically  amounts  to  no  more  than  this,  that  of  the  regular 
solids,  with  all  sides  plane  and  similar,  this  one  has  the  least  surface 
for  its  solid  content. 

The  rhombic  dodecahedron  has  six  tetrahedral  angles,  and 
eight  trihedral  angles ;  and  it  is  obvious,  on  consideration,  that 
at  each  of  the  former  six  dodecahedra  meet  in  a  point,  and  that, 
where  the  four  tetrahedral  facets  of  each  coalesce  with  their 
neighbours,  we  have  twelve  plane  films,  or  interfaces,  meeting  in 
a  point.  In  a  precisely  similar  fashion,  we  may  imagine  twelve 
plane  films,  drawn  inwards  from  the  twelve  edges  of  a  cube,  to 
meet  at  a  point  in  the  centre  of  the  cube.  But,  as  Plateau  dis- 
covered*, when  we  dip  a  cubical 
wire  skeleton  into  soap-solution  and 
take  it  out  again,  the  twelve  films 
which  are  thus  generated  do  not 
meet  in  a  point,  but  are  grouped 
around  a  small  central,  plane,  quadri- 
lateral film  (Fig.  134).  In  other 
words,  twelve  plane  films,  meeting  in 
a  point,  are  essentially  unstable.  If 
we  blow  upon  our  artificial  film- 
system,  the  little  quadrilateral  alters 
its  place,  setting  itself  parallel  now  to  one  and  now  to  another  of 
the  paired  faces  of  the  cube ;  but  we  never  get  rid  of  it.  Moreover, 
the  size  and  shape  of  the  quadrilateral,  as  of  all  the  other  films  in  the 
system,  are  perfectly  definite.     Of  the  twelve  films  (which  we  had 

*  Also  discovered  independently  by  Sir  David  Brewster,  Trans.  R.S.E.  xxiv, 
p.  505,  1867,  XXV,  p.  115,  1869. 

T.  G.  22 


Fig.  134. 


338  THE   FORMS   OF  TISSUES  [ch. 

expected  to  find  all  plane  and  all  similar)  four  afe  plane  isosceles 
triangles,  and  eight  are  slightly  curved  quadrilateral  figures.  The 
former  have  two  curved  sides,  meeting  at  an  angle  of  109°  28', 
and  their  apices  coincide  with  the  corners  of  the  central  quadri- 
lateral, whose  sides  are  also  curved,  and  also  meet  at  this  identical 
angle ; — which  (as  we  observe)  is  likewise  an  angle  which  we  have 
been  dealing  with  in  the  simpler  case  of  the  bee's  cell,  and  indeed 
in  all  the  regular  solids  of  which  we  have  yet  treated. 

By  completing  the  assemblage  of  polyhedra  of  which  Plateau's 
skeleton-cube  gives  a  part.  Lord  Kelvin  shewed  that  we  should 
obtain  a  set  of  equal  and  similar  fourteen-sided  figures,  or  "  tetra- 
kaidecahedra " ;  and  that  by  means  of  an  assemblage  of  these 
figures  space  is  homogeneously  partitioned — that  is  to  say,  into 
equal,  similar  and  similarly  situated  cells — with  an  economy  of 
surface  in  relation  to  area  even  greater  than  in  an  assemblage  of 
rhombic  dodecahedra. 

In  the  most  generalised  case,  the  tetrakaidecahedron  is  bounded 
by  three  pairs  of  equal  and  parallel  quadrilateral  faces,  and  four 
pairs  of  equal  and  parallel  hexagonal  faces,  neither  the  quadri- 
laterals nor  the  hexagons  being  necessarily  plane.  In  a  certain 
particular  case,  the  quadrilaterals  are  plane  surfaces,  but  the 
hexagons  shghtly  curved  "anticlastic''  surfaces;  and  these  latter 
have  at  every  point  equal  and  opposite  curvatures,  and  are 
surfaces  of  minimal  curvature  for  a  boundary  of  six  curved  edges. 
The  figure  has  the  remarkable  property  that,  like  the  plane 
rhombic  dodecahedron,  it  so  partitions  space  that  three  faces 
meeting  in  an  edge  do  so  everywhere  at  equal  angles  of  120°*. 

We  may  take  it  as  certain  that,  in  a  system  of  perfectly  fluid 
films,  like  the  interior  of  a  mass  of  soap-bubbles,  where  the  films 
are  perfectly  free  to  glide  or  to  rotate  over  one  another,  the  mass 
is  actually  divided  into  cells  of  this  remarkable  conformation. 

*  Von  Fedorow  had  already  described  (in  Russian)  the  same  figure,  under  the 
name  of  cubo-octahedron,  or  hejita-parallelohedron,  Umited  however  to  the  case 
where  all  the  faces  are  plane.  This  figure,  together  with  the  cube,  the  hexagonal 
prism,  the  rhombic  dodecahedron  and  the  "elongated  dodecahedron,"  constituted 
the  five  plane-faced,  parallel-sided  figures  by  which  space  is  capable  of  being 
completely  filled  and  symmetrically  partitioned ;  the  series  so  forming  the  founda- 
tion of  Von  Fedorow's  theory  of  crystaUine  structure.  The  elongated  dodecahedron 
is,  essentiaUj^  the  figure  of  the  bee's  cell. 


VII]  OF   THE   PARTITIONING   OF   SPACE  339 

And  it  is  quite  possible,  also,  that  in  the  cells  of  a  vegetable 
parenchyma,  by  carefully  macerating  them  apart,  the  same  con- 
formation may  yet  be  demonstrated  under  suitable  conditions ; 
that  is  to  say  when  the  whole  tissue  is  highly  symmetrical,  and  the 
individual  cells  are  as  nearly  as  possible  equal  in  size.  But  in  an 
ordinary  microscopic  section,  it  would  seem  practically  impossible 
to  distinguish  the  fourteen-sided  figure  from  the  twelve-sided. 
Moreover,  if  we  have  anything  whatsoever  interposed  so  as  to 
prevent  our  twelve  films  meeting  in  a  point,  and  (so  to  speak)  to 
take  the  place  of  our  little  central  quadrilateral, — if  we  have,  for 
instance,  a  tiny  bead  or  droplet  in  the  centre  of  our  artificial 
system,  or  even  a  little  thickening,  or  "  bourrelet"  as  Plateau  called 
it,  of  the  cell-wall,  then  it  is  no  longer  necessary  that  the 
tetrakaidecahedron  should  be  formed.  Accordingly,  it  is  very 
probably  the  case  that,  in  the  parenchymatous  tissue,  under  the 
actual  conditions  of  restraint  and  of  very  imperfect  fluidity,  it  is 
after  all  the  rhombic  dodecahedral  configuration  which,  even  under 
perfectly  symmetrical  conditions,  is  generally  assumed. 

It  follows  from  all  that  we  have  said,  that  the  problems 
connected  with  the  conformation  of  cells,  and  with  the  manner  in 
which  a  given  space  is  partitioned  by  them,  soon  become  exceedingly 
complex.  And  while  this  is  so  even  when  all  our  cells  are  equal 
and  symmetrically  placed,  it  becomes  vastly  more  so  when  cells 
varying  even  slightly  in  size,  in  hardness,  rigidity  or  other  quahties, 
are  packed  together.  The  mathematics  of  the  case  very  soon 
become  too  hard  for  us ;  but  in  its  essence,  the  phenomenon 
remains  the  same.  We  have  little  reason  to  doubt,  and  no  just 
cause  to  disbeheve,  that  the  whole  configuration,  for  instancef  of 
an  egg  in  the  advanced  stages  of  segmentation,  is  accurately 
determined  by  simple  physical  laws,  just  as  much  as  in  the  early 
stages  of  two  or  four  cells,  during  which  early  stages  we  are  able  to 
recognise  and  demonstrate  the  forces  and  their  resultant  effects. 
But  when  mathematical  investigation  has  become  too  difiicult,  it 
often  happens  that  physical  experiment  can  reproduce  for  us  the 
phenomena  which  Nature  exhibits  to  us,  and  which  we  are  striving 
to  comprehend.  For  instance,  in  an  admirable  research,  M.  Robert 
shewed,  some  years  ago,  not  only  that  the  early  segmentation  of 

22—2 


340  THE  FOKMS   OF   TISSUES  [ch. 

the  egg  of  Trochus  (a  marine  univalve  mollusc)  proceeded  in 
accordance  with  the  laws  of  surface  tension,  but  he  also  succeeded 
in  imitating  by  means  of  soap-bubbles,  several  stages,  one  after 
another,  of  the  developing  egg. 

M.  Robert  carried  his  experiments  as  far  as  the  stage  of 
sixteen  cells,  or  bubbles.  It  is  not  easy  to  carry  the  artificial 
system  quite  so  far,  but  in  the  earlier  stages  the  experiment  is 
easy ;  we  have  merely  to  blow  our  bubbles  in  a  little  dish,  adding 
one  to  another,  and  adjusting  their  sizes  to  produce  a  symmetrical 
system.  One  of  the  simplest  and  prettiest  parts  of  his  investigation 
concerned  t^ie  "polar  furrow"  of  which  we  have  spoken  on  p.  310. 
On  blowing  four  little  contiguous  bubbles  he  found  (as  we  may 
all  find  with  the  greatest  ease)  that  they  form  a  symmetrical  system, 
two  in  contact  with  one  another  by  a  laminar  film,  and  two, 
which  are  elevated  a  little  above  the  others,  and  which  are  separated 
by  the  length  of  the  aforesaid  lamina.  The  bubbles  are  thus  in 
contact  three  by  three,  their  partition-walls  making  with  one 
another  equal  angles  of  120°.  The  upper  and  lower  edges  of  the 
intermediate  lamina  (the  lower  one  visible  through  the  transparent 
system)  constitute  the  two  polar  furrows  of  the  embryologist 
(Fig.  135,  1-3).  The  lamina  itself  is  plane  when  the  system  is 
symmetrical,  but  it  responds  by  a  corresponding  curvature  to 
the  least  inequality  of  the  bubbles  on  either  side.  In  the 
experiment,  the  upper  polar  furrow  is  usually  a  little  shorter 
than  the  lower,  but  parallel  to  it;  that  is  to  say,  the  lamina 
is  of  trapezoidal  form:  this  lack  of  perfect  symmetry  being 
due  (in  the  experimental  case)  to  the  lower  portion  of  the 
bubbles  being  somewhat  drawn  asunder  by  the  tension  of  their 
at|B.chments  to  the  sides  of  the  dish  (Fig.  135,  4).  A  similar 
phenomenon  is  usually  found  in  Trochus,  according  to  Robert, 
and  many  other  observers  have  likewise  found  the  upper  furrow 
to  be  shorter  than  the  one  below.  In  the  various  species  of  the 
genus  Crepidula,  Conklin  asserts  that  the  two  furrows  are  equal 
in  C.  convexa,  that  the  upper  one  is  the  shorter  in  C.  fornicata, 
and  that  the  upper  one  all  but  disappears  in  C.  plana ;  but  we  may 
well  be  permitted  to  doubt,  without  the  evidence  of  very  special 
investigations,  whether  these  slight  physical  differences  are 
actually  characteristic  of,  and  constant  in,  particular  allied  species. 


VIl] 


OF   POLAR  FURROWS 


341 


Returning  to  the  experimental  case,  Robert  found  that  by  with- 
drawing a  Uttle  air  from,  and  so  diminishing  the  bulk  of  the  two 
terminal  bubbles  (i.e.  those  at  the  ends  of  the  intermediate  lamina), 
the  upper  polar  furrow  was  caused  to  elongate,  till  it  became  equal 
in  length  to  the  lower;  and  by  continuing  the  process  it  became 
the  longer  in  its  turn.     These  two  conditions  have  again  been 


Fig.   135.     Aggregations  of  four  soap-bubbles,  to  shew  various  arrangements  of 
the  intermediate  partition  and  polar  furrows.     (After  Robert.) 


described  by  investigators  as  characteristic  of  this  embryo  or  that ; 
for  instance  in  Unio,  Lillie  has  described  the  two  furrows  as 
gradually  altering  their  respective  lengths* ;  and  Wilson  (as  Lillie 
remarks)  had  already  pointed  out  that  "the  reduction  of  the 
apical  cross-furrow,  as  compared  with  that  at  the  vegetative  pole 

*  F.  R.  Lillie,  Embryology  of  the  Unionidae,  Journ.  of  Morphology,  x,  p.  12, 
1895. 


342  THE  FORMS   OF  TISSUES  [ch. 

in  molluscs  and  annelids  '  stands  in  obvious  relation  to  the  different 
size  of  the  cells  produced  at  the  two  poles*.'" 

When  the  two  lateral  bubbles  are  gradually  reduced  in  size, 
or  the  two  terminal  ones  enlarged,  the  upper  furrow  becomes 
shorter  and  shorter;  and  at  the  moment  when  it  is  about  to 
vanish,  a  new  furrow  makes  its  instantaneous  appearance  in  a 
direction  perpendicular  to  the  old  one;  but  the  inferior  furrow, 
constrained  by  its  attachment  to  the  base,  remains  unchanged, 
and  accordingly  our  two  polar  furrows,  which  were  formerly 
parallel,  are  now  at  right  angles  to  one  another.  Instead  of  a 
single  plane  quadrilateral  partition,  we  have  now  two  triangular 
ones,  meeting  in  the  middle  of  the  system  by  their  apices,  and 
lying  in  planes  at  right  angles  to  one  another  (Fig.  135,  5-7) f. 
Two  such  polar  furrows,  equal  in  length  and  arranged  in  a  cross, 
have  again  been  frequently  described  by  the  embryologists. 
Robert  himself  found  this  condition  in  Trochus,  as  an  occasional 
or  exceptional  occurrence:  it  has  been  described  as  normal  in 
Asterina  by  Ludwig,  in  Branchipus  by  Spangenberg,  and  in 
Podocoryne  and  Hydractinia  by  Bunting.  It  is  evident  that  it 
represents  a  state  of  unstable  equilibrium,  only  to  be  maintained 
under  certain  conditions  of  restraint  within  the  system. 

So,  by  sUght  and  delicate  modifications  in  the  relative  size  of 
the  cells,  we  may  pass  through  all  the  possible  arrangements  of  the 
median  partition,  and  of  the  "furrows"  which  correspond  to  its 
upper  and  lower  edges ;  and  every  one  of  these  arrangements  has 
been  frequently  observed  in  the  four-celled  stage  of  various  embryos. 
As  the  phases  pass  one  into  the  other,  they  are  accompanied  by 
changes  in  the  curvature  of  the  partition,  which  in  like  manner 
correspond  precisely  to  phenomena  which  the  embryologists  have 
witnessed  and  described.  And  all  these  configurations  belong  to 
that  large  class  of  phenomena  whose  distribution  among  embryos, 
or  among  organisms  in  general,  bears  no  relation  to  the  boundaries 
of  zoological  classification ;    through  molluscs,  worms,  coelenter- 

*  E.  B.  Wilson,  The  Cell-lineage  of  Nereis,  Journ.  of  Morphology,  vi,  p.  452, 
1892. 

t  It  is  highly  probable,  and  we  may  reasonably  assume,  that  the  two  little 
triangles  do  not  actually  meet  at  an  apical  point,  but  merge  into  one  another  by 
a  twist,  or  minute  surface  of  complex  curvature,  so  as  not  to  contravene  the  normal 
conditions  of  equihbrium. 


vii]  OF  POLAR  FURROWS  343 

ates,  vertebrates  and  what  not,  we  meet  with  now  one  and  now 
another,  in  a  medley  which  defies  classification.  They  are  not 
"vital  phenomena,"  or  "functions"  of  the  organism,  or  special 
characteristics  of  this  or  that  organism,  but  purely  physical 
phenomena.  The  kindred  but  more  complicated  phenomena 
which  correspond  to  the  polar  furrow  when  a  larger  number  of 
cells  than  four  are  associated  together,  we  shall  deal  with  in  the 
next  chapter. 

Having  shewn  that  the  capillary  phenomena  are  patent  and 
unmistakable  during  the  earlier  stages  of  embryonic  development, 
but  soon  become  more  obscure  and  incapable  of  experimental 
reproduction  in  the  later  stages,  when  the  cells  have  increased  in 
number,  various  writers  including  Robert  himself  have  been 
inclined  to  argue  that  the  physical  phenomena  die  away,  and  are 
overpowered  and  cancelled  by  agencies  of  a  very  different  order. 
Here  we  pass  into  a  region  where  direct  observation  and  experi- 
ment are  not  at  hand  to  guide  us,  and  where  a  man's  trend  of 
thought,  and  way  of  judging  the  whole  evidence  in  the  case,  must 
shape  his  philosophy.  We  must  remember  that,  even  in  a  froth 
of  soap-bubbles,  we  can  apply  an  exact  analysis  only  to  the  simplest 
cases  and  conditions  of  the  phenomenon ;  we  cannot  describe, 
but  can  only  imagine,  the  forces  which  in  such  a  froth  control  the 
respective  sizes,  positions  and  curvatures  of  the  innumerable 
bubbles  and  films  of  which  it  consists ;  but  our  knowledge  is 
enough  to  leave  us  assured  that  what  we  have  learned  by  in- 
vestigation of  the  simplest  cases  includes  the  principles  which 
determine  the  most  complex.  In  the  case  of  the  growing  embryo 
we  know  from  the  beginning  that  surface  tension  is  only  one  of 
the  physical  forces  at  work;  and  that  other  forces,  including 
those  displayed  within  the  interior  of  each  living  cell,  play  their 
part  in  the  determination  of  the  system.  But  we  have  no  evidence 
whatsoever  that  at  this  point,  or  that  point,  or  at  any,  the  dominion 
of  the  physical  forces  over  the  material  system  gives  place  to  a 
new  condition  where  agencies  at  present  unknown  to  the  physicist 
impose  themselves  on  the  living  matter,  and  become  responsible 
for  the  conformation  of  its  material  fabric. 

Before  we  leave  for  the  present  the  subject  of  the  segmenting 


344  THE   FORMS   OF   TISSUES  [ch. 

egg,  we  must  take  brief  note  of  two  associated  problems:  viz. 
(1)  the  formation  and  enlargement  of  the  segmentation  cavity,  or 
central  interspace  around  which  the  cells  tend  to  group  themselves 
in  a  single  layer,  and  (2)  the  formation  of  the  gastrula,  that  is  to 
say  (in  a  typical  case)  the  conversion  "by  invagination,"  of  the 
one-layered  ball  into  a  two-layered  cup.  Neither  problem  is  free 
from  difficulty,  and  all  we  can  do  meanwhile  is  to  state  them  in 
general  terms,  introducing  some  more  or  less  plausible  assumptions. 

The  former  problem  is  comparatively  easy,  as  regards  the 
tendency  of  a  segmentation  cavity  to  enlarge,  when  once  it  has 
been  established.  We  may  then  assume  that  subdivision  of  the 
cells  is  due  to  the  appearance  of  a  new-formed  septum  within  each 
cell,  that  this  septum  has  a  tendency  to  shrink  under  surface 
tension,  and  that  these  changes  will  be  accompanied  on  the  whole 
by  a  diminution  of  surface  energy  in  the  system.  This  being  so, 
it  may  be  shewn  that  the  volume  of  the  divided  cells  must  be  less 
than  it  was  prior  to  division,  or  in  other  words  that  part  of  their 
contents  must  exude  during  the  process  of  segmentation*. 
Accordingly,  the  case  where  the  segmentation  cavity  enlarges  and 
the  embryo  developes  into  a  hollow  blastosphere  may,  under  the 
circumstances,  be  simply  described  as  the  case  where  that  outflow 
or  exudation  from  the  cells  of  the  blastoderm  is  directed  on  the 
whole  inwards. 

The  physical  forces  involved  in  the  invagination  of  the  cell- 
layer  to  form  the  gastrula  have  been  repeatedly  discussed  f,  but 
the  true  explanation  seems  as  yet  to  be  by  no  means  clear.  The 
case,  however,  is  probably  not  a  very  difficult  one,  provided  that 
we  may  assume  a  difference  of  osmotic  pressure  at  the  two  poles 
of  the  blastosphere,  that  is  to  say  between  the  cells  which  are 
being  differentiated  into  outer  and  inner,  into  epiblast  and  hypo- 
blast. It  is  plain  that  a  blastosphere,  or  hollow  vesicle  bounded 
by  a  layer  of  vesicles,  is  under  very  different  physical  conditions 
from  a  single,  simple  vesicle  or  bubble.  The  blastosphere  has  no 
effective  surface  tension  of  its  own,  such  as  to  exert  pressure  on 

*  Professor  Peddie  has  given  me  this  interesting  and  important  result,  but  the 
mathematical  reasoning  is  too  lengthy  to  be  set  forth  here. 

t  Cf.  Rhumbler,  Arch.  f.  Entw.  Mech.  xrv,  p.  401,  1902;  Assheton,  ibid,  xxxi, 
pp.  46-78,  1910. 


VII]  OR  CELL-AGGREGATES  345 

its  contents  or  bring  the  whole  into  a  spherical  form ;  nor  will  local 
variations  of  surface  energy  be  directly  capable  of  affecting  the 
form  of  the  system.  But  if  the  substance  of  our  blastosphere  be 
sufficiently  viscous,  then  osmotic  forces  may  set  up  currents 
which,  reacting  on  the  external  fluid  pressure,  may  easily  cause 
modifications  of  shape;  and  the  particular  case  of  invagination 
itself  will  not  be  difficult  to  account  for  on  this  assumption  of 
non-uniform  exudation  and  imbibition. 


CHAPTER  VIII 

THE   FORMS   OF  TISSUES   OR   CELL-AGGREGATES  (continued) 

The  problems  which  we  have  been  considering,  and  especially 
that  of  the  bee's  cell,  belong  to  a  class  of  "  isoperimetrical " 
problems,  which  deal  with  figures  whose  surface  is  a  minimum  for 
a  definite  content  or  volume.  Such  problems  soon  become 
difficult,  but  we  may  find  many  easy  examples  which  lead  us 
towards  the  explanation  of  biological  phenomena ;  and  the 
particular  subject  which  we  shall  find  most  easy  of  approach  is 
that  of  the  division,  in  definite  proportions,  of  some  definite 
portion  of  space,  by  a  partition-wall  of  minimal  area.  The 
theoretical  principles  so  arrived  at  we  shall  then  attempt  to  apply, 
after  the  manner  of  Berthold  and  Errera,  to  the  actual  biological 
phenomena  of  cell-division. 

This  investigation  we  may  approach  in  two  ways :  by  con- 
sidering, namely,  the  partitioning  ofi  from  some  given  space  or 
area  of  one-half  (or  some  other  fraction)  of  its  content ;  or  again, 
by  dealing  simultaneously  with  the  partitions  necessary  for  the 
breaking  up  of  a  given  space  into  a  definite  number  of  compart- 
ments. 

If  we  take,  to  begin  with,  the  simple  case  of  a  cubical  cell,  it 
is  obvious  that,  to  divide  it  into  two  halves,  the  smallest  possible 
partition-wall  is  one  which  runs  parallel  to,  and  midway  between, 
two  of  its  opposite  sides.  If  we  call  a  the  length  of  one  of  the 
edges  of  the  cube,  then  a^  is  the  area,  alike  of  one  of  its  sides,  and 
of  the  partition  which  we  have  interposed  parallel,  or  normal, 
thereto.  But  if  we  now  consider  the  bisected  cube,  and  wish  to 
divide  the  one-half  of  it  again,  it  is  obvious  that  another  partition 
parallel  to  the  first,  so  far  from  being  the  smallest  possible,  is 
precisely  twice  the  size  of  a  cross-partition  perpendicular  to  it; 


CH.  VIIl] 


OF  SACHS'S  EULES 


347 


for  the  area  of  this  new  partition  is  a  x  a/2.  And  again,  for  a 
third  bisection,  our  next  partition  must  be  perpendicular  to  the 
other  two,  and  it  is  obviously  a  little  square,  with  an  area  of 

From  this  we  may  draw  the  simple  rule  that,  for  a  rectangular 
body  or  parallelopiped  to  be  divided  equally  by  means  of  a 
partition  of  minimal  area,  (1)  the  partition  must  cut  across  the 
longest  axis  of  the  figure ;  and  (2)  in  the  event  of  successive 
bisections,  each  partition  must  run  at  right  angles  to  its  immediate 
predecessor. 


a 


/\  ^^y% 

/ 

/ 

/ 

/ 

/ 

---. 

/ 

Fig.  136.     (After  Berthold.) 


We  have  already  spoken  of  "Sachs's  Rules,"  which  are  an 
empirical  statement  of  the  method  of  cell-division  in  plant- tissues ; 
and  we  may  now  set  them  forth  in  full. 

(1)  The  cell  typically  tends  to  divide  into  two  co-equal  parts. 

(2)  Each  new  plane  of  division  tends  to  intersect  at  right 
angles  the  preceding  plane  of  division. 

The  first  of  these  rules  is  a  statement  of  physiological  fact, 
not  without  its  exceptions,  but  so  generally  true  that  it  will 
justify  us  in  hmiting  our  enquiry,  for  the  most  part,  to  cases  of 
equal  subdivision.  That  it  is  by  no  means  universally  true  for 
cells  generally  is  shewn,  for  instance,  by  such  well-known  cases 


348  THE   FORMS   OF  TISSUES  [ch. 

as  the  unequal  segmentation  of  the  frog's  egg.  It  is  true  when  the 
dividing  cell  is  homogeneous,  and  under  the  influence  of  symmetrical 
forces  ;  but  it  ceases  to  be  true  when  the  field  is  no  longer  dynami- 
cally symmetrical,  for  instance,  when  the  parts  difier  in  surface 
tension  or  internal  pressure.  This  latter  condition,  of  asymmetry 
of  field,  is  frequent  in  segmenting  eggs*,  and  is  then  equivalent 
to  the  principle  upon  which  Balfour  laid  stress,  as  leading  to 
"unequal"  or  to  "partial"  segmentation  of  the  egg, — viz.  the 
unequal  or  asymmetrical  distribution  of  protoplasm  and  of  food- 
yolk. 

The  second  rule,  which  also  has  its  exceptions,  is  true  in  a 
large  number  of  cases;  and  it  owes  its  validity,  as  we  may  judge 
from  the  illustration  of  the  repeatedly  bisected  cube,  solely  to  the 
guiding  principle  of  minimal  areas.  It  is  in  short  subordinate 
to,  and  covers  certain  cases  included  under,  a  much  more  important 
and  fundamental  rule,  due  not  to  Sachs  but  to  Errera  ;  that  (3)  the 
incipient  partition- wall  of  a  dividing  cell  tends  to  be  such  that  its 
area  is  the  least  possible  by  which  the  given  space-content  can  be 
enclosed. 

Let  us  return  to  the  case  of  our  cube,  and  let  us  suppose  that, 
instead  of  bisecting  it,  we  desire  to  shut  off  some  small  portion 
only  of  its  volume.  It  is  found  in  the  course  of  experiments  upon 
soap-films,  that  if  we  try  to  bring  a  partition-film  too  near  to  one 
side  of  a  cubical  (or  rectangular)  space,  it  becomes  unstable ;  and 
is  easily  shifted  to  a  totally  new  position,  in  which  it  constitutes 
a  curved  cylindrical  wall,  cutting  off  one  corner  of  the  cube. 
It  meets  the  sides  of  the  cube  at  right  angles  (for  reasons  which  we 
have  already  considered) ;   and,  as  we  may  see  from  the  symmetry 

*  M.  Robert  {I.  c.  p.  305)  has  compiled  a  long  list  of  cases  among  the  molluscs 
and  the  worms,  where  the  initial  segmentation  of  the  egg  proceeds  by  equal  or 
unequal  division.  The  two  cases  are  about  equally  numerous.  But  like  many 
other  writers,  he  would  ascribe  this  equaUty  or  inequahty  rather  to  a  provision 
for  the  future  than  to  a  direct  effect  of  immediate  physical  causation :  "  li  semble 
assez  probable,  comm-3  on  I'a  dit  souvent,  que  la  plus  grande  taille  d'un  blastomere 
est  liee  a  I'importance  et  au  developpement  precoce  des  parties  du  corps  qui  doivent 
en  naitre :  il  y  aurait  la  une  sorte  de  reflet  des  stades  posterieures  du  developpement 
sur  les  premieres  phenomenes,  ce  que  M.  Ray  Lankester  appelle  precocious  segrega- 
tion. II  faut  avouer  pourtant  qu'on  est  parfois  assez  embarrasse  pour  assignor  une 
cause  a  pareilles  differences." 


VIIl] 


OF  AREAE   MINIMAE 


349 


of  the  case,  it  constitutes  precisely  one-quarter  of  a  cylinder. 
Our  plane  transverse  partition,  wherever  it  was  placed,  had  always 
the  same  area,  viz.  a^;  and  it  is  obvious  that  a  cylindrical  wall, 
if  it  cut  ofE  a  small  corner,  may  be  much  less  than  this.  We  want, 
accordingly,  to  determine  what  is  the  particular  volume  which 
might  be  partitioned  off  with  equal  economy  of  wall-space  in  one 
way  as  the  other,  that  is  to  say,  what  area  of  cylindrical  wall 
would  be  neither  more  nor  less  than  the  area  a^.  The  calculation 
is  very  easy. 

The  surface-area  of  a  cylinder  of  length  a  is  277r  .  a,  and  that 
of  our  quarter-cylinder  is,  therefore,  a  .  7rr/2 ;  and  this  being,  by 
hypothesis,  =  a'^,  we  have  a  =  7Tr/2,  or  r  =  2a/7r. 

The  volume  of  a  cylinder,  of  length  a,  is  a-Trr^,  and  that  of  our 
quarter-cylinder  is  a  .  -n-r^ji,  which  (by  substituting  the  value  of  r) 
is  equal  to  a^/Tr. 

Now  precisely  this  same  volume  is,  obviously,  shut  off  by  a 
transverse  partition  of  area  a^,  if  the  third  side  of  the  rectangular 
space  be  equal  to  a/ir.  And  this  fraction,  if  we  take  a  =  1,  is 
equal  to  0-318... ,  or  rather  less  than  one-third.  And,  as  we  have 
just  seen,  the  radius,  or  side,  of  the  corresponding  quarter-cylinder 
will  be  twice  that  fraction,  or  equal  to  -636  times  the  side  of  the 
cubical  cell. 

If  then,  in  the  process  of  division 
of  a  cubical  cell,  it  so  divide  that  the 
two  portions  be  not  equal  in  volume 
but  that  one  portion  by  anything  less 
than  about  three-tenths  of  the  whole, 
or  three-sevenths  of  the  other  portion, 
there  will  be  a  tendency  for  the  cell 
to  divide,  not  by  means  of  a  plane 
transverse  partition,  but  by  means  of 
a  curved,  cylindrical  wall  cutting  off 
one  corner  of  the  original  cell ;  and 
the  part  so  cut  off  will  be  one-quarter  of  a  cylinder. 

By  a  similar  calculation  we  can  shew  that  a  spherical  wall, 
cutting  off  one  solid  angle  of  the  cube,  and  constituting  an  octant 
of  a  sphere,  would  likewise  be  of  less  area  than  a  plane  partition 
as  soon  as  the  volume  to  be  enclosed  was  not  greater  than  about 


Fig.  137. 


350 


THE  FOKMS   OF   TISSUES 


[CH. 


one-quarter  of  the  original  cell*.  But  while  both  the  cyhndrical 
wall  and  the  spherical  wall  would  be  of  less  area  than  the  plane 
transverse  partition  after  that  limit  (of  one-quarter  volume)  was 
passed,  the  cyhndrical  would  still  be  the  better  of  the  two  up  to 
a  further  limit.     It  is  only  when  the  volume  to  be  partitioned  ofE 


*  The  principle  is  well  illustrated  in  an  experiment  of  Sir  David  Brewster's 
{Trans.  R.8.E.  xxv,  p.  Ill,  1869).  A  soap-film  is  drawn  over  the  rim  of  a  wine- 
glass, and  then  covered  by  a  watch-glass.  The  film  is  inclined  or  shaken  tiU  it 
becomes  attached  to  the  glass  covering,  and  it  then  immediately  changes  place, 
leaving  its  transverse  position  to  take  up  that  of  a  spherical  segment  extending 
from  one  side  of  the  wine-glass  to  its  cover,  and  so  enclosing  the  same  volume  of 
air  as  formerly  but  with  a  great  economy  of  surface,  precisely  as  in  the  case  of  our 
spherical  partition  cutting  off  one  corner  of  a  cube. 


VIII]  OF   AREAE   MINIMAE  351 

is  no  greater  than  about  0-15,  or  somewhere  about  one-seventh, 
of  the  whole/ that  the  spherical  cell- wall  in  an  angle  of  the  cubical 
cell,  that  is  to  say  the  octant  of  a  sphere,  is  definitely  of  less  area 
than  the  quarter-cylinder.  In  the  accompanying  diagram  (Fig.  138) 
the  relative  areas  of  the  three  partitions  are  shewn  for  all  fractions, 
less  than  one-half,  of  the  divided  cell. 

In  this  figure,  we  see  that  the  plane  transverse  partition,  whatever  fraction 
of  the  cube  it  cut  off,  is  always  of  the  same  dimensions,  that  is  to  say  is 
always  equal  to  a^,  or  =  1.  If  one-half  of  the  cube  have  to  be  cut  off,  this 
plane  transverse  partition  is  much  the  best,  for  we  see  by  the  diagram  that  a 
cylindrical  partition  cutting  off  an  equal  volume  would  have  an  area  about 
25%,  and  a  spherical  partition  would  have  an  area  about  50%  greater. 
The  point  A  in  the  diagram  corresponds  to  the  point  where  the  cyhndrical 
partition  would  begin  to  have  an  advantage  over  the  plane,  that  is  to  say 
(as  we  have  seen)  when  the  fraction  to  be  cut  off  is  about  one-thiixl,  or  -318 
of  the  whole.  In  like  manner,  at  B  the  spherical  octant  begins  to  have  an 
advantage  over  the  plane ;  and  it  is  not  till  we  reach  the  point  C  that  the 
spherical  octant  becomes  of  less  area  than  the  quarter- cylinder. 

The  case  we  have  dealt  with  is  of  little  practical  importance  to 
the  biologist,  because  the  cases  in  which 
a  cubical,  or  rectangular,  cell  divides 
unequally,  and  unsymmetrically,  are 
apparently  few;  but  we  can  find,  as 
Berthold  pointed  out,  a  few  examples, 
for  instance  in  the  hairs  within  the 
reproductive  "conceptacles"  of  certain 
Fuci  (Sphacelaria,  etc..  Fig.  139),  or  in 
the  "paraphyses"  of  mosses  (Fig.  142). 
But  it  is  of  great  theoretical  importance ;  as  serving  to  introduce 
us  to  a  large  class  of  cases,  in  which  the  shape  and  the  relative 
dimensions  of  the  original  cavity  lead,  according  to  the  principle 
of  minimal  areas,  to  cell-division  in  very  definite  and  sometimes 
unexpected  ways.  It  is  not  easy,  nor  indeed  possible,  to  give  a 
generalised  account  of  these  cases,  for  the  limiting  conditions 
are  somewhat  complex,  and  the  mathematical  treatment  soon 
becomes  difficult.  But  it  is  easy  to  comprehend  a  few  simple 
cases,  which  of  themselves  will  carry  us  a  good  long  way;  and 
which  will  go  far  to  convince  the  student  that,  in  other  cases 


352  THE  FORMS   OF  TISSUES  [ch. 

which  we  cannot  fully  master,  the  same  guiding  principle  is  at 
the  root  of  the  matter. 


The  bisection  of  a  solid  (or  the  subdivision  of  its  volume  in 
other  definite  proportions)  soon  leads  us  into  a  geometry  which, 
if  not  necessarily  difficult,  is  apt  to  be  unfamiliar;  but  in  such 
problems  we  can  go  a  long  way,  and  often  far  enough  for  our 
particular  purpose,  if  we  merely  consider  the  plane  geometry  of 
a  side  or  section  of  our  figure.  For  instance,  in  the  case  of  the 
cube  which  we  have  been  just  considering,  and  in  the  case  of  the 
plane  and  cyHndrical  partitions  by  which  it  has  been  divided,  it 
is  obvious  that,  since  these  two  partitions  extend  symmetrically 
from  top  to  bottom  of  our  cube,  that  we  need  only  consider  (so 
far  as  they  are  concerned)  the  manner  in  which  they  subdivide 
the  base  of  the  cube.  The  whole  problem  of  the  solid,  up  to  a 
certain  point,  is  contained  in  our  plane  diagram  of  Fig.  138.  And 
when  our  particular  sohd  is  a  solid  of  revolution,  then  it  is  obvious 
that  a  study  of  its  plane  of  symmetry  (that  is  to  say  any  plane 
passing  through  its  axis  of  rotation)  gives  us  the  solution  of  the 
whole  problem.  The  right  cone  is  a  case  in  point,  for  here  the 
investigation  of  its  modes  of  symmetrical  subdivision  is  completely 
met  by  an  examination  of  the  isosceles  triangle  which  constitutes 
its  plane  of  symmetry. 

The  bisection  of  an  isosceles  triangle  by  a  line  which  shall 
be  the  shortest  possible  is  a  very  easy  problem.  Let  ABC  be 
such  a  triangle  of  which  A  is  the  apex ;  it  may  be  shewn  that, 
for  its  shortest  line  of  bisection,  we  are  limited  to  three  cases : 
viz.  to  a  vertical  line  AD,  bisecting  the  angle  at  A  and  the  side 
BC ;  to  a  transverse  line  parallel  to  the  base  BC ;  or  to  an  oblique 
line  parallel  to  AB  or  to  AC.  The  respective  magnitudes,  or 
lengths,  of  these  partition  lines  follow  at  once  from  the  magnitudes 
of  the  angles  of  our  triangle.  For  we  know,  to  begin  with,  since 
the  areas  of  similar  figures  vary  as  the  squares  of  their  linear 
dimensions,  that,  in  order  to  bisect  the  area,  a  line  parallel  to  one 
side  of  our  triangle  must  always  have  a  length  equal  to  l/v2 
of  that  side.  If  then,  we  take  our  base,  BC,  in  all  cases  of 
a  length  =  2,  the  transverse  partition  drawn  parallel  to  it  will 
always  have  a   length   equal  to  2/-\/2,  or   =  y/'I.     The  vertical 


VIIl] 


OF  AREAE   MINIMAE 


353 


partition,  AD,  since  BD  =  1,  will  always  equal  tan^  (/3  being 
the  angle  ABC).  And  the  oblique  partition,  GH,  being  equal  to 
AB/\/'2=^  l/-V2co&  ^.      If  then  we  call    our  vertical,  transverse 


and  oblique  partitions,  V,  T,  and  0,  we  have  F  =  tan  j8 ; 
T  =  V2  ;    and  0  =  1/a/2  cos  j8,  or 

V  -.T  :0  =  tan^/V2  :  1  :  1/2  cos  ^. 

And,  working  out  these  equations  for  various  values  of  ^,  we 
very  soon  see  that  the  vertical  partition  (F)  is  the  least  of  the 
three  until  j8  =  45°,  at  which  limit  F  and  0  are  each  equal  to 
1/V2  =  -707 ;  and  that  again,  when  ^  =  60°,  0  and  T  are  each 
=  1,  after  which  T  (whose  value  always  =  1)  is  the  shortest  of 
the  three  partitions.  And,  as  we  have  seen,  these  results  are  at 
once  appHcable,  not  only  to  the  case  of  the  plane  triangle,  but 
also  to  that  of  the  conical  cell. 


Fig.  141. 

In  like  manner,  if  we  have  a  spheroidal  body,  less  than 
a  hemisphere,  such  for  instance  as  a  low,  watch-glass  shaped 
cell  (Fig.  141,  a),  it  is  obvious  that  the  smallest  possible 
partition    by    which    we    can    divide   it   into   two    equal    halves 

T.  G.  23 


354  THE  FORMS   OF  TISSUES  [ch. 

is    (as    in    our    flattened    disc)    a    median    vertical    one.      And 
likewise,   the   hemisphere   itself   can   be   bisected   by  no   smaller 
partition   meeting   the  walls   at   right   angles  than  that  median 
one  which  divides  it  into  two  similar  quadrants  of   a  sphere. 
But  if  we  produce  our  hemisphere  into  a  more  elevated,  conical 
body,  or  into  a  cylinder  with  spherical  cap,  it  is  obvious  that  there 
comes  a  point  where  a  transverse,  horizontal  partition  will  bisect 
the  figure  with  less  area  of  partition-wall  than  a  median  vertical 
one  (c).     And  furthermore,  there  will  be  an  intermediate  region, 
a  region  where  height  and  base  have  their  relative  dimensions 
nearly  equal  (as  in  b),  where  an  obhque  partition  will  be  better 
than  either  the  vertical  or  the  transverse,  though  here  the  analogy 
of  our  triangle  does  not  suffice  to  give  us  the  precise  limiting 
values.     We  need  not  examine  these  limitations  in  detail,  but  we 
must  look  at  the  curvatures  which  accompany  the  several  con- 
ditions.    We  have  seen  that  a  film  tends  to  set  itself  at  equal 
angles  to  the  surface  which  it  meets,  and  therefore,  when  that 
surface  is  a  solid,  to  meet  it  (or  its  tangent  if  it  be  a  curved  surface) 
at  right  angles.     Our  vertical  partition  is,  therefore,  everywhere 
normal  to  the  original  cell- walls,  and  constitutes  a  plane  surface. 
But  in  the  taller,  conical  cell  with  transverse  partition,  the 
latter  still  meets  the  opposite  sides  of  the  cell  at  right  angles,  and 
it  follows  that  it  must  itself  be  curved;    moreover,  since  the 
tension,  and  therefore  the  curvature,  of  the  partition  is  every- 
where uniform,  it  follows  that  its  curved  surface  must  be  a  portion 
of  a  sphere,  concave  towards  the  apex  of  the  original,  now  divided, 
cell.     In  the  intermediate  case,  where  we  have  an  oblique  partition, 
meeting  both  the  base  and  the  curved  sides  of  the  mother- cell, 
the  contact  must  still  be  everywhere  at  right  angles :    provided 
we  continue  to  suppose  that  the  walls  of  the  mother-cell  (like  those 
of  our  diagrammatic  cube)  have  become  practically  rigid  before 
the  partition  appears,  and  are  therefore  not  affected  and  deformed 
by  the  tension  of  the  latter.     In  such  a  case,  and  especially  when 
the  cell  is  elhptical  in  cross-section,  or  is  still  more  compHcated 
in  form,  it  is  evident  that  the  partition,  in  adapting  itself  to 
circumstances  and  in  maintaining  itself  as  a  surface  of  minimal 
area  subject  to  all  the  conditions  of  the  case,  may  have  to  assume 
a  complex  curvature. 


VIII]       OF  SIGMOID   OR   S-SHAPED   PARTITIONS         355 


While  in  very  many  cases  the  partitions  (hke  the  walls  of  the 
original  cell)  will  be  either  plane  or  spherical,  a  more  complex 
curvature  will  be  assumed  under  a  variety  of  conditions.  It  will 
be  apt  to  occur,  for  instance,  when  the  mother-cell  is  irregular  in 
shape,  and  one  particular  case  of  such  asymmetry  will  be  that  in 
which  (as  in  Fig.  143)  the  cell  has  begun  to  brahch,  or  give  ofE  a 
diverticulum,  before  division  takes  place.  A  very  complicated 
case  of  a  different  kind,  though  not  without  its  analogies  to  the 
cases  we  are  considering,  will  occur  in  the  partitions  of  minimal 
area  which  subdivide  the  spiral  tube  of  a  nautilus,  as  we  shall 


C  D 

Fig.  142.  S-shaped  partitions:  A,  from  Taonia  atomaria  (after  Reinke);  B,  from 
paraphyses  of  Fucus;  C,  from  rhizoids  of  Mossj  D,  from  paraphyses  of 
Polytrichum. 

presently  see.  And  again,  whenever  we  have  a  marked  internal 
asymmetry  of  the  cell,  leading  to  irregular  and  anomalous  modes 
of  division,  in  which  the  cell  is  not  necessarily  divided  into  two 
equal  halves  and  in  which  the  partition-wall  may  assume  an 
oblique  position,  then  apparently  anomalous  curvatures  will  tend 
to  make  their  appearance*. 

Suppose  that  a  more  or  less  oblong  cell  have  a  tendency  to 

divide  by  means  of  an  oblique  partition  (as  may  happen  through 

various  causes  or  conditions  of  asymmetry),  such  a  partition  will 

still  have  a  tendency  to  set  itself  at  right  angles  to  the  rigid  walls 

*  Cf.  Wildeman,  Attache  des  Clo  sons,  etc.,  pis.  1,  2. 

23—2 


356 


THE   FORMS   OF   TISSUES 


[CH. 


of  the  mother-cell :  and  it  will  at  once  follow  that  our  oblique 
partition,  throughout  its  whole  extent,  will  assume  the  form  of 
a  complex,  saddle-shaped  or  anticlastic  surface. 

Many  such  cases  of  partitions  with  complex  or  double  curvature 
exist,  but  they  are  not  always  easy  of  recognition,  nor  is  the 
particular  case  where  they  appear  in  a  terminal  cell  a  common 
one.  We  may  see  them,  for  instance,  in  the  roots  (or  rhizoids) 
of  Mosses,  especially  at  the  point  of  development  of  a  hew  rootlet 
(Fig.  142,  C) ;  and  again  among  Mosses,  in  the  "paraphyses"  of 
the  male  prothalli  (e.g.  in  Polytrichum),  we  find  more  or  less 
similar  partitions  (D).  They  are  frequent  also  among  many  Fuci, 
as  in  the  hairs  or  paraphyses  of  Fucus  itself  (B).     In  Taonia 


" 


A  B  CD 

Fig.  143.     Diagrammatic  explanation  of  S-shaped  partition. 


atomaria,  as  figured  in  Reinke's  memoir  on  the  Dictyotaceae  of 
the  Gulf  of  Naples*,  we  see,  in  like  manner,  oblique  partitions, 
which  on  more  careful  examination  are  seen  to  be  curves  of 
double  curvature  (Fig.  142,  A). 

The  physical  cause  and  origin  of  these  S-shaped  partitions  is 
somewhat  obscure,  but  we  may  attempt  a  tentative  explanation. 
When  we  assert  a  tendency  for  the  cell  to  divide  transversely  to 
its  long  axis,  we  are  not  only  stating  empirically  that  the  partition 
tends  to  appear  in  a  small,  rather  than  a  large  cross-section  of  the 
cell :  but  we  are  also  implicitly  ascribing  to  the  cell  a  longitudinal 
polarity  (Fig.   143,  A),  and  implicitly  asserting  that  it  tends  to 


*  Nova  Acta  K.  Leop.  Akad.  si,  1,  pi.  iv. 


VIII]       OF  SIGMOID   OR  S-SHAPED   PARTITIONS         357 

divide  (just  as  the  segmenting  egg  does),  by  a  partition  transverse 
to  its  polar  axis.  Such  a  polarity  may  conceivably  be  due  to 
a  chemical  asvmmetry,  or  anisotropy,  such  as  we  have  learned 
of  (from  Professor  Macallum's  experiments)  in  our  chapter  on 
Adsorption.  Now  if  the  chemical  concentration,  on  which  this 
anisotropy  or  polarity  (by  hypothesis)  depends,  be  unsymmetrical, 
one  of  its  poles  being  as  it  were  deflected  to  one  side,  where  a  little 
branch  or  bud  is  being  (or  about  to  be)  given  off, — all  in  precise 
accordance  with  the  adsorption  phenomena  described  on  p.  289, — 
then  our  "polar  axis"  would  necessarily  be  a  curved  axis,  and  the 
partition,  being  constrained  (again  ex  hyjpoihesi)  to  arise  transversely 
to  the  polar  axis,  would  lie  obliquely  to  the  apparent  axis  of  the 
cell  (Fig.  143,  B,  C).  And  if  the  obhque  partition  be  so  situated 
that  it  has  to  meet  the  opposite  walls  (as  in  C),  then,  in  order  to 
do  so  symmetrically  (i.e.  either  perpendicularly,  as  when  the 
cell-wall  is  already  sohdified,  or  at  least  at  equal  angles  on  either 
side),  it  is  evident  that  the  partition,  in  its  course  from  one  side 
of  the  cell  to  the  other,  must  necessarily  assume  a  more  or  less 
S-shaped  curvature  (Fig.  143,  D). 

As  a  matter  of  fact,  while  we  have  abundant  simple  illustrations 
of  the  principles  which  we  have  now  begun  to  study,  apparent 
exceptions  to  this  simplicity,  due  to  an  asymmetry  of  the  cell 
itself,  or  of  the  system  of  which  the  single  cell  is  but  a  part,  are 
by  no  means  rare.  For  example,  we  know  that  in  cambium-cells, 
division  frequently  takes  place  parallel  to  the  long  axis  of  the 
cell,  when  a  partition  of  much  less  area  would  suffice  if  it  were 
set  cross-ways :  and  it  is  only  when  a  considerable  disproportion 
has  been  set  up  between  the  length  and  breadth  of  the  cell,  that 
the  balance  is  in  part  redressed  by  the  appearance  of  a  transverse 
partition.  It  was  owing  to  such  exceptions  that  Berthold  was 
led  to  qualify  and  even  to  depreciate  the  importance  of  the  law 
of  minimal  areas  as  a  factor  in  cell-division,  after  he  himself  had 
done  so  much  to  demonstrate  and  elucidate  it*.  He  was  deeply 
and  rightly  impressed  by  the  fact  that  other  forces  besides  surface 

*  Cf.  Protoplasmamechanik,  p.  229:  "Insofern  liegen  also  die  Verhaltnisse  hier 
wesentlich  anders  als  bei  der  Zertheiluno;  hohler  Korperformen  durch  fliissige 
Lamellen.  Wenn  die  Membran  bei  der  Zelltheilung  die  von  dem  Prinzip  der 
kleinsten  Flachen  geforderte  Lage  und  Kriimmung  annimmt,  so  werden  wir  den 
Grund  dafiir  in  andrer  Weise  abzuleiten  haben." 


358  THE  FORMS   OF   TISSUES  [ch. 

tension^  both  external  and  internal  to  the  cell,  play  their  part 
in  the  determination  of  its  partitions,  and  that  the  answer  to 
our  problem  is  not  to  be  given  in  a  word.  How  fundamentally 
important  it  is,  however,  in  spite  of  all  conflicting  tendencies  and 
apparent  exceptions,  we  shall  see  better  and  better  as  we  proceed. 

But  let  us  leave  the  exceptions  and  return  to  a  consideration 
of  the  simpler  and  more  general  phenomena.  And  in  so  doing, 
let  us  leave  the  case  of  the  cubical,  quadrangular  or  cylindrical 
cell,  and  examine  the  case  of  a  spherical  cell  and  of  its  successive 
divisions,  or  the  still  simpler  case  of  a  circular,  discoidal  cell. 

When  we  attempt  to  investigate  mathematically  the  position 
and  form  of  a  partition  of  minimal  area,  it  is  plain  that  we  shall 
be  dealing  with  comparatively  simple  cases  wherever  even  one 
dimension  of  the  cell  is  much  less  than  the  other  two.  Where  two 
dimensions  are  small  compared  with  the  third,  as  in  a  thin  cylin- 
drical filament  Uke  that  of  Spirogyra,  we  have  the  problem  at  its 
simplest;  for  it  is  at  once  obvious,  then,  that  the  partition  must 
lie  transversely  to  the  long  axis  of  the  thread.  But  even  where 
one  dimension  only  is  relatively  small,  as  for  instance  in  a  flattened 
plate,  our  problem  is  so  far  simplified  that  we  see  at  once  that  the 
partition  cannot  be  parallel  to  the  extended  plane,  but  must  cut 
the  cell,  somehow,  at  right  angles  to  that  plane.  In  short,  the 
problem  of  dividing  a  much  flattened  solid  becomes  identical  with 
that  of  dividing  a  simple  surface  of  the  same  form. 

There  are  a  number  of  small  Algae,  growing  in  the  form  of 
small  flattened  discs,  consisting  (for  a  time  at  any  rate)  of  but  a 
single  layer  of  cells,  which,  as  Berthold  shewed,  exemplify  this 
comparatively  simple  problem;  and  we  shall  find  presently  that 
it  is  also  admirably  illustrated  in  the  cell-divisions  which  occur  in 
the  egg  of  a  frog  or  a  sea-urchin,  when  the  egg  for  the  sake  of 
experiment  is  flattened  out  under  artificial  pressure. 

Fig.  144  (taken  from  Berthold's  Monograph  of  the  Naples 
Bangiaciae)  represents  younger  and  older  discs  of  the  little  alga 
Erythrotrichia  discigera ;  and  it  will  be  seen  that,  in  all  stages  save 
the  first,  we  have  an  arrangement  of  cell-partitions  which  looks 
somewhat  complex,  but  into  which  we  must  attempt  to  throw  some 
light  and  order.     Starting  with  the  original  single,  and  flattened. 


VIIl] 


THE   SEGMENTATION   OF  A  DISC 


359 


cell,  we  have  no  difficulty  with  the  first  two  cell-divisions;  for 
we  know  that  no  bisecting  partitions  can  possibly  be  shorter  than 
the  two  diameters,  which  divide  the  cell  into  halves  and  into 


Fig.   144.     Development  of  Erythrotrichia.     (After  Berthold. 


quarters.  We  have  only  to  remember  that,  for  the  sum  total  of 
partitions  to  be  a  minimum,  three  only  must  meet  in  a  point; 
and  therefore,  the  'four  quadrantal  walls  must  shift  a  httle,  pro- 
ducing the  usual  httle  median  partition,  or  cross-furrow,  instead 
of  one  common,  central  point  of  junction.  This  little  inter- 
mediate wall,  however,  will  be  very  small,  and  to  all  intents  and 
purposes  we  may  deal  with  the 
case  as  though  we  had  now  to  do 
with  four  ec[ual  cells,  each  one  of 
them  a  perfect  quadrant.  And 
so  our  problem  is,  to  find  the 
shortest  line  which  shall  divide  the 
quadrant  of  a  circle  into  two 
halves  of  equal  area.  A  radial 
partition  (Fig.  145,  a),  starting 
from  the  apex  of  the  quadrant,  is 
at  once  excluded,  for  a  reason 
similar  to  that  just  referred  to; 
our  choice  must  lie  therefore  between  two  modes  of  division  such 
as  are  illustrated  in  Fig.  145,  where  the  partition  is  either  (as  in  b) 


Fig.   14.5. 


360  THE   FORMS   OF   TISSUES  [ch. 

concentric  with  the  outer  border  of  the  cell,  or  else  (as  in  c)  cuts 
that  outer  border;  in  other  words,  our  partition  may  (b)  cut  both 
radial  walls,  or  (c)  may  cut  one  radial  wall  and  the  periphery. 
These  are  the  two  methods  of  division  which  Sachs  called,  respec- 
tively, (b)  periclinal,  and  (c)  anticlinal*.  We  may  either  treat  the 
walls  of  the  dividing  quadrant  as  already  solidified,  or  at  least  as 
having  a  tension  compared  with  which  that  of  the  incipient 
partition  film  is  inconsiderable.  In  either  case  the  partition  must 
meet  the  cell-wall,  on  either  side,  at  right  angles,  and  (its  own 
tension  and  curvature  being  everywhere  uniform)  it  must  take  the 
form  of  a  circular  arc. 

Now  we  find  that  a  flattened  cell  which  is  approximately  a 
quadrant  of  a  circle  invariably  divides  after  the  manner  of 
Fig.  145,  c,  that  is  to  say,  by  an  approximately  circular,  anticlinal 
wall,  such  as  we  now  recognise  in  the  eight-celled  stage  of 
Erythrotrichia  (Fig.  144) ;  let  us  then  consider  that  Nature  has 
solved  our  problem  for  us,  and  let  us  work  out  the  actual 
geometric  conditions. 

Let  the  quadrant  OAB  {in  Fig.  146)  be  divided  into  two 
parts  of  equal  area,  by  the  circular  arc  MP.  It  is  required  to 
determine  (1)  the  position  of  P  upon  the  arc  of  the  quadrant, 
that  is  to  say  the  angle  BOP ;  (2)  the  position  of  the  point  M 
on  the  side  OA ;  and  (3)  the  length  of  the  arc  MP  in  terms  of  a 
radius  of  the  quadrant. 

(1)  Draw  OP;  also  PC  a  tangent,  meeting  OA  in  C;  and 
PN,  perpendicular  to  OA.  Let  us  call  a  a  radius ;  and  6  the  angle 
at  C,  which  is  obviously  equal  to  OPN,  or  POB.     Then 

CP  =  a  cot  d;  PN  =  a  cos  6;  NC  =  CP  cos  6  =  a  .  cos^  ^/sin  d. 

The  area  of  the  portion  PMN 

=  \CP'-  e  -  IPN  .  NC 

=  \a^  cot^  6  —  la  cos  d  .  a  cos^  6 /sin  6 

=  \a^  (cot2  d  -  cos3  djsm  6). 

*  There  is,  I  tliink,  some  ambiguity  or  disagreement  among  botanists  as  to  the 
use  of  this  latter  term :  the  sense  in  which  I  am  using  it,  viz.  for  any  partition 
which  meets  the  outer  or  peripheral  wall  at  right  angles  (the  strictly  radial  partition 
being  for  the  present  excluded),  is,  however,  clear. 


VIIl] 


THE   BISECTION  OF   A  QUADRANT 


361 


And  the  area  of  the  portion  PNA 

=  1^2  (77/2  -6)-  ION  .  NP 
=  ^a^  {ttI2  —  6)  —  |a  sin  ^  .  a  cos  6 
=  |a2  (77/2  -  d-&me  .  cos  9). 
Therefore  the  area  of  the  whole  portion  PMA 
=  a2/2  (7r/2  -6+6  cot^  6  -  cos^  dj&m  6  -  sin  6  .  cos  6) 
=  a''/2  (7r/2  -6+6  cot^  6  -  cot  6), 
and  also,  by  hypothesis,  =  f .  area  of  the  quadrant,  =  Tra^/S. 


Fig.   14(3. 

Hence  6  is  defined  by  the  equation 

a^/2  (77/2  -  ^  +  0  cot2  6  -  cot  0)  =  77a2/8, 
or  TTJi-  6  +  6  cot^  ^  -  cot  6*  =  0. 

We  may  solve  this  equation  by  constructing  a  table  (of  which 
the  following  is  a  small  portion)  for  various  values  of  6. 


e 

./4 

-e 

-cote 

+  d  cot2  e 

=  x 

34°  34' 

•7854  - 

•6033  - 

-  1-4514 

+  1-2709  - 

-0016 

35' 

•7854 

■6036 

1-4505 

1-2700 

-0013 

36' 

•7851 

•6039 

1-4496 

1-2690 

-0009 

37' 

•7854 

•6042 

b4487 

1-2680 

-0005 

38' 

•7854 

•6045 

1-4478 

1-2671 

-0002 

39' 

•7854 

•6048 

1-4469 

1-2661     - 

-  -0002 

40' 

•7854 

•6051 

1-4460 

1-2652 

-  -0005 

362  THE   FORMS   OF   TISSUES  [cii. 

We  see  accordingly  that  the  equation  is  solved  (as  accurately 
as  need  be)  when  6  is  an  angle  somewhat  over  34°  38'^  or  say 
34°  38|'.  That  is  to  say,  a  quadrant  of  a  circle  is  bisected  by  a 
circular  arc  cutting  the  side  and  the  periphery  of  the  quadrant 
at  right  angles,  when  the  arc  is  such  as  to  include  (90°  —  34°  38'), 
i.e.  55°  22'  of  the  quadrantal  arc. 

This  determination  of  ours  is  practically  identical  with  that 
which  Berthold  arrived  at  by  a  rough  and  ready  method,  without 
the  use  of  mathematics.  He  simply  tried  various  ways  of  dividing 
a  quadrant  of  paper  by  means  of  a  circular  arc,  and  went  on  doing 
so  till  he  got  the  weights  of  his  two  pieces  of  paper  approximately 
equal.  The  angle,  as  he  thus  determined  it,  was  34-6°,  or  say 
34°  36'. 

(2)  The  position  of  M  on  the  side  of  the  quadrant  OA  is 
given  by  the  equation  OM  =  a  cosec  6  —  a  cot  6 ;  the  value  of 
which  expression,  for  the  angle  which  we  have  just  discovered, 
is  -3028.  That  is  to  say,  the  radius  (or  side)  of  the  quadrant  will 
be  divided  by  the  new  partition  into  two  parts,  in  the  proportions 
of  nearly  three  to  seven. 

(3)  The  length  of  the  arc  MP  is  equal  to  ad  cot  6 ;  and  the 
value  of  this  for  the  given  angle  is  •8751.  This  is  as  much  as  to 
say  that  the  curved  partition- wall  which  we  are  considering  is 
shorter  than  a  radial  partition  in  the  proportion  of  8f  to  10,  or 
seven-eights  almost  exactly. 

But  we  must  also  compare  the  length  of  this  curved  "  antichnal " 
partition- wall  {MP)  with  that  of  the  con- 
centric, or  periclinal,  one  {RS,  Fig.  147)  by 
which  the  quadrant  might  also  be  bisected. 
The  length  of  this  partition  is  obviously 
equal  to  the  arc  of  the  quadrant  (i.e.  the 
peripheral  wall  of  the  cell)  divided  by  v'2 ; 
or,  in  terms  of  the  radius,  =  7r/2V2  =  1-111. 
^^*  So  that,  not  only  is  the  anticlinal  partition 

(such  as  we  actually  find  in  nature)  notably  the  best,  but  the 
periclinal  one,  when  it  comes  to  dividing  an  entire  quadrant,  is 
very  considerably  larger  even  than  a  radial  partition. 

The  two  cells  into  which  our  original  quadrant  is  now  divided, 
while  they  are  equal  in  volume,  are  of  very  different  shapes ;   the 


VIIl] 


THE   BISECTION   OF   A    QUADRANT 


363 


one  is  a  triangle  (MAP)  with  two  sides  formed  of  circular  arcs, 
and  the  other  is  a  four-sided  figure  (MOBP),  which  we  may  call 
approximately  oblong.  We  cannot  say  as  yet  how  the  triangular 
portion  ought  to  divide ;  but  it  is  obvious  that  the  least  possible 
partition-wall  which  shall  bisect  the  other  must  run  across  the 
long  axis  of  the  oblong,  that  is  to  say  periclinally.  This,  also,  is 
precisely  what  tends  actually  to  take  place.  In  the  following 
diagrams  (Fig.  148)  of  a  frog's  egg  dividing  under  pressure,  that 
is  to  say  when  reduced  to  the  form  of  a  flattened  plate,  we  see, 
firstly,  the  division  into  four  quadrants  (by  the  partitions  1,  2) ; 
secondly,  the  division  of  each  quadrant  by  means  of  an  anti- 
clinal circular  arc  (3,  3),  cutting  the  peripheral  wall  of  the  quadrant 
approximately  in  the  proportions  of  three  to  seven ;    and  thirdly. 


Fig.  148.     Segmentation  of  frog's  egg,  under  artificial  compression. 
(After  Roux.) 

we  see  that  of  the  eight  cells  (four  triangular  and  four  oblong) 
into  which  the  whole  egg  is  now  divided,  the  four  which  we  have 
called  oblong  now  proceed  to  divide  by  partitions  transverse  to 
their  long  axes,  or  roughly  parallel  to  the  periphery  of  the  egg. 


The  question  how  the  other,  or  triangular,  portion  of  the  divided 
qudarant  will  next  divide  leads  us  to  another  well-defined  problem, 
which  is  only  a  slight  extension,  making  allowance  for  the  circular 
arcs,  of  that  elementary  problem  of  the  triangle  we  have  already 
considered.  We  know  now  that  an  entire  quadrant  must  divide 
(so  that  its  bisecting  wall  shall  have  the  least  possible  area)  by 
means  of  an  antichnal  partition,  but  how  about  any  smaller 
sectors  of  circles?     It  is  obvious  in  the  case  of  a  small  prismatic 


364  ■        THE   FORMS   OF   TISSUES  [ch. 

sector,  such  as  that  shewn  in  Fig.  149,  that  a  periclinal  partition 
is  the  smallest  by  which  we  can  possibly  bisect  the  cell ;  we  want, 
accordingly,  to  know  the  limits  below  which  the  perichnal  partition 
is  always  the  best,  and  above  which  the  anticlinal  arc,  as  in  the 
case  of  the  whole  quadrant,  has  the  advantage  in  regard  to  small- 
ness  of  surface  area. 

This  may  be  easily  determined ;  for  the  preceding  investigation 
is  a  perfectly  general  one,  and  the  results  hold  good  for  sectors 
of  any  other  arc,  as  well  as  for  the  quadrant,  or  arc  of  90°.  That 
is  to  say,  the  length  of  the  partition- wall  MP  is  always  determined 
by  the  angle  6,  according  to  our  equation  MP  =  ad  cot  6 ;  and 
the  angle  6  has  a  definite  relation  to  a,  the  angle  of  arc. 


OA-yfO. 


Fig.  149. 

Moreover,  in  the  case  of  the  periclinal  boundary,  RS  (Fig.  147) 
{or  ab,  Fig.  149),  we  know  that,  if  it  bisect  the  cell, 

RS  =  a  .  a/V2. 
Accordingly,  the  arc  RS  will  be  just  equal  to  the  arc  MP  when 

d  cot  6  =  a/V2. 

When  ^  cot  ^  >  a/V2,  or  MP  >  RS, 

then  division  will  take  place  as  in  RS. 

When  6  cot  6  <  a/V2,  or  MP  <  RS, 

then  division  will  take  place  as  in  MP. 

In  the  accompanying  diagram  (Fig.  150),  I  have  plotted  the 
various  magnitudes  with  which  we  are  concerned,  in  order  to 
exhibit  the  several  limiting  values.  Here  we  see,  in  the  first 
place,  the  curve  marked  a,  which  shews  on  the  (left-hand)  vertical 
scale  the  various  possible  magnitudes  of  that  angle  (viz.  the  angle 


VIIl] 


THE   BISECTION  OF  A   QUADRANT 


365 


of  arc  of  the  whole  sector  which  we  wish  to  di\dde),  and  on  the 
horizontal  scale  the  corresponding  values  of  6,  or  the  angle  which 

Angle  (6)  determining  the  intersection  of  the  partition -wall  with  the  outer  border 

of  the  cell. 


2-0 
1-8 

w 

3 

eS 

ft 

1-4  ^ 

M 

a 

Hl-2l 

a 

aT 

o 
1-0  .-s 

-  .8    S 


ft 

0" 

20° 

30° 

40° 

50 

60°              70°              80° 

90 

ts 

\^ 

1 

1 

( 

1                   1 

o 

'M 

^, 

\ 

yD' 

^  80 

- 

\ 

■I 

C3            O 

- 

'\ 

\cx 

- 

,'« 

\ 

\ 

F' 

g    60 

- 

^ 

^ 

d 

\^ 

o 

w 

\ 

B 

"^       Kf? 

- 

c' 

.2    50 

a 

D 

\ 

\                        .<>*/ 

/ 

&            0 

«    40 

-c 

'o 

2  3d 

3 

'^^ 

E 

F' 

F' 

- 

o 
U  2d 

id 

~ 

^^i 

J 

F'" 

~-^     \  ^>^ 

- 

^^■'^"'^ 

'•""'^ 

X,,^         \ 

0 

'    1 

1            1            1 

1            1            1-  "" 

\ 

-•2 


Fig.  150. 


determines  the  point  on  the  periphery  where  it  is  cut  by  the 
partition- wall,  MP.  Two  limiting  cases  are  to  be  noticed  here: 
(1)  at  90°  (point  A  in  diagram),  because  we  are  at  present  only 


366  THE  FORMS   OF  TISSUES  [ch. 

dealing  with  arcs  no  greater  than  a  quadrant;  and  (2),  the  point 
{B)  where  the  angle  6  comes  to  equal  the  angle  a,  for  after  that 
point  the  construction  becomes  impossible,  since  an  anticlinal 
bisecting  partition- wall  would  be  partly  outside  the  cell.  The  only 
partition  which,  after  the  point,  can  possibly  exist,  is  a  periclinal 
one.  This  point,  as  our  diagram  shews  us,  occurs  when  the  angles 
(a  and  6)  are  each  rather  under  52°. 

Next  I  have  plotted,  on  the  same  diagram,  and  in  relation  to 
the  same  scales  of  angles,  the  corresponding  lengths  of  the  two 
partitions,  viz.  RS  and  MP,  their  lengths  being  expressed  (on 
the  right-hand  side  of  the  diagram)  in  relation  to  the  radius  of 
the  circle  (a),  that  is  to  say  the  side  wall,  OA,  of  our  cell. 

The  limiting  values  here  are  (1),  C,  C ,  where  the  angle  of  arc 
is  90°,  and  where,  as  we  have  already  seen,  the  two  partition- walls 
have  the  relative  magnitudes  of  MP  :  RS  =  0-875  :  1-111 ;  (2)  the 
point  D,  where  RS  equals  unity,  that  is  to  say  where  the  periclinal 
partition  has  the  same  length  as  a  radial  one;  this  occurs  when 
a  is  rather  under  82°  (cf.  the  points  Z),  D');  (3)  the  point  E,  where 
RS  and  MP  intersect ;  that  is  to  say  the  point  at  which  the  two 
partitions,  periclinal  and  anticHnal,  are  of  the  same  magnitude; 
this  is  the  case,  according  to  our  diagram,  when  the  angle  of  arc 
is  just  over  62|°.  We  see  from  this,  then,  that  what  we  have 
called  an  anticlinal  partition,  as  MP,  is  only  hkely  to  occur  in 
a  triangular  or  prismatic  cell  whose  angle  of  arc  lies  between 
90°  and  62|°.  In  all  narrower  or  more  tapering  cells,  the  periclinal 
partition  will  be  of  less  area,  and  will  therefore  be  more  and  more 
likely  to  occur. 

The  case  {F)  where  the  angle  a  is  just  60°  is  of  some  interest. 
Here,  owing  to  the  curvature  of  the  peripheral  border,  and  the 
consequent  fact  that  the  peripheral  angles  are  somewhat  greater 
than  the  apical  angle  a,  the  perichnal  partition  has  a  very  shght 
and  almost  imperceptible  advantage  over  the  anticlinal,  the 
relative  proportions  being  about  as  MP  :  RS  =  0-73  :  0-72?  But  if 
the  equilateral  triangle  be  a  plane  spherical  triangle,  i.e.  a  plane 
triangle  bounded  by  circular  arcs,  then  we  see  that  there  is  no 
longer  any  distinction  at  all  between  our  two  partitions;  MP 
and  RS  are  now  identical. 

On  the  same  diagram,  I  have  inserted  the  curve  for  values  of 


VIIl] 


THE   SEGMENTATION   OF  A  DISC 


367 


cosec  6  —  cot  6  =  OM,  that  is  to  say  the  distances  from  the  centre, 
along  the  side  of  the  cell,  of  the  starting-point  (M)  of  the  anticlinal 
partition.  The  point  C"  represents  its  position  in  the  case  of 
a  quadrant,  and  shews  it  to  be  (as  we  have  already  said)  about 
3/10  of  the  length  of  the  radius  from  the  centre.  If,  on  the  other 
hand,  our  cell  be  an  equilateral  triangle,  then  we  have  to  read  off 
the  point  on  this  curve  corresponding  to  a  =  60°,  and  we  find  it 
at  the  point  F'"  (vertically  under  F),  which  tells  us  that  the 
partition  now  starts  4-5/10,  or  nearly  halfway,  along  the  radial 
wall. 


The  foregoing  considerations  carry  us  a  long  way  in  our 
investigations  of  many  of  the  simpler  forms  of  cell-division. 
Strictly  speaking  they  are  limited  to  the  case  of  flattened  cells, 
in  which  we  can  treat  the  problem  as  though  we  were  simply 
partitioning  a  plane  surface.  But  it  is  obvious  that,  though  they 
do  not  teach  us  the  whole  conformation  of  the  partition  which 
divides  a  more  compUcated  solid  into  two  halves,  yet  they  do,  even 
in  such  a  case,  enlighten  us  so  far,  that  they  tell  us  the  appearance 
presented  in  one  plane  of  the  actual  solid.  And  as  this  is  all  that 
we  see  in  a  microscopic  section,  it  follows  that  the  results  we  have 
arrived  at  will  greatly  help  us  in  the  interpretation  of  microscopic 
appearances,  even  in  comparatively  complex  cases  of  cell-division. 

Let  us  now  return  to  our 
quadrant  cell  {OAPB),  which  we 
have  found  to  be  divided  into 
a  triangular  and  a  quadrilateral 
portion,  as  in  Fig.  147  or  Fig.  151 ; 
and  let  us  now  suppose  the  whole 
system  to  grow,  in  a  uniform 
fashion,  as  a  prelude  to  further 
subdivision.  The  whole  quadrant, 
growing  uniformly  (or  with  equal 
radial  increments),  will  still  re- 
main a  quadrant,  and  it  is 
obvious,  therefore,  that  for  every  -^     j^j 

new  increment  of  size,  more  will 
be  added  to  the  margin  of  its  triangular  portion  than  to  the 


368  THE  FOKMS   OF  TISSUES  [ch. 

narrower  margin  of  its  quadrilateral  portion;  and  these  incre- 
ments will  be  in  proportion  to  the  angles  of  arc,  viz.  55°  22' :  34°  38', 
or  as  '96  :  -60,  i.e.  as  8  :  5.  And  accordingly,  if  we  may  assume 
(and  the  assumption  is  a  very  plausible  one),  that,  just  as  the 
quadrant  itself  divided  into  two  halves  after  it  got  to  a  certain 
size,  so  each  of  its  two  halves  will  reach  the  same  size  before 
again  dividing,  it  is  obvious  that  the  triangular  portion  will  be 
doubled  in  size,  and  therefore  ready  to  divide,  a  considerable 
time  before  the  quadrilateral  part.  To  work  out  the  problem  in 
detail  would  lead  us  into  troublesome  mathematics ;  but  if 
we  simply  assume  that  the  increments  are  proportional  to  the 
increasing  radii  of  the  circle,  we  have  the  following  equations :- — 

Let  us  call  the  triangular  cell  T,  and  the  quadrilateral,  Q 
(Fig.  151) ;  let  the  radius,  OA,  of  the  original  quadrantal  cell 
=  a  =  1 ;  and  let  the  increment  which  is  required  to  add  on  a 
portion  equal  to  T  (such  as  PP'A'A)  be  called  x,  and  let  that 
required,  similarly,  for  the  doubling  of  Q  be  called  x'. 

Then  we  see  that  the  area  of  the  original  quadrant 

^T  +Q  =  l7ra2  =  .7854a^ 
while  the  area  of  T  ^Q=  •S927a^. 

The  area  of  the  enlarged  sector,  p'OA', 

=  {a  +  xY  X  (55^  22')  -^  2  =  -4831  (a  +  xf, 
and  the  area  OP  A 

=  a2  X  (55°  22')  -^  2  =  •4831a2. 

Therefore  the  area  of  the  added  portion,  T', 

=  -4831{(a  +  a;)2-a2}. 

And  this,  by  hypothesis, 

=  T  =  •3927a2. 

We  get,  accordingly,  since  a  =  1, 

a:2  +  2a;  -  •3927/-4831  =  -810, 
and,  solving, 

a;  +  1  =  VbSi  =  1-345,  or  a;  -  0-345. 

Working  out  x'  in  the  same  way,  we  arrive  at  the  approximate 
value,  a;'  +  1  =  1-517. 


VIII]  THE   SEGMENTATION   OF   A  DISC  369 

This  is  as  much  as  to  say  that,  supposing  each  cell  tends  to 
divide  into  two  halves  when  (and  not  before)  its  original  size  is 
doubled,  then,  in  our  flattened  disc,  the  triangular  cell  T  will  tend 
to  divide  when  the  radius  of  the  disc  has  increased  by  about  a 
third  (from  1  to  1-345),  but  the  quadrilateral  cell,  Q,  will  not  tend 
to  divide  until  the  linear  dimensions  of  the  disc  have  increased 
by  about  a  half  (from  1  to  1-517). 

The  case  here  illustrated  is  of  no  small  general  importance. 
For  it  shews  us  that  a  uniform  and  symmetrical  growth  of  the 
organism  (symmetrical,  that  is  to  say,  under  the  limitations  of  a 
plane  surface,  or  plane  section)  by  no  means  involves  a  uniform 
or  symmetrical  growth  of  the  individual  cells,  but  may,  under 
certain  conditions,  actually  lead  to  inequality  among  these;  and 
this  inequality  may  be  further  emphasised  by  differences  which 
arise  out  of  it,  in  regard  to  the  order  of  frequency  of  further 
subdivision.  This  phenomenon  (or  to  be  quite  candid,  this 
hypothesis,  which  is  due  to  Berthold)  is  entirely  independent  of 
any  change  or  variation  in  individual  surface  tensions ;  and 
accordingly  it  is  essentially  different  from  the  phenomenon  of 
unequal  segmentation  (as  studied  by  Balfour),  to  which  we  have 
referred  on  p.  348. 

In  this  fashion,  we  might  go  on  to  consider  the  manner,  and 
the  order  of  succession,  in  which  the  subsequent  cell-divisions 
would  tend  to  take  place,  as  governed  by  the  principle  of  minimal 
areas.  But  the  calculations  would  grow  more  diSicult,  or  the 
results  got  by  simple  methods  would  grow  less  and  less  exact. 
At  the  same  time,  some  of  these  results  would  be  of  great  interest, 
and  well  worth  the  trouble  of  obtaining.  For  instance,  the  precise 
manner  in  which  our  triangular  cell,  T,  would  next  divide  would 
be  interesting  to  know,  and  a  general  solution  of  this  problem  is 
certainly  troublesome  to  calculate.  But  in  this  particular  case 
we  can  see  that  the  width  of  the  triangular  cell  near  P  is  so 
obviously  less  than  that  near  either  of  the  other  two  angles,  that 
a  circular  arc  cutting  off  that  angle  is  bound  to  be  the  shortest 
possible  bisecting  line;  and  that,  in  short,  our  triangular  cell 
will  tend  to  subdivide,  just  like  the  original  quadrant,  into  a 
triangular  and  a  quadrilateral  portion. 

But  the  case  will  be  different  next  time,  because  in  this  new 

T.  G.  24 


370 


THE  FORMS  OF  TISSUES 


[CH. 


triangle,  PRQ,  the  least  width  is  near  the  innermost  angle,  that 
at  Q ;  and  the  bisecting  circular  arc  will  therefore  be  opposite  to  Q, 
or  (approximately)  parallel  to  PR.  The  importance  of  this  fact  is 
at  once  evident;  for  it  means  to  say  that  there  soon  comes  a 
time  when,  whether  by  the  division  of  triangles  or  of  quadrilaterals, 
we  find  only  quadrilateral  cells  adjoining  the  periphery  of  our 
circular  disc.  In  the  subsequent  division  of  these  quadrilaterals, 
the  partitions  will  arise  transversely  to  their  long  axes,  that  is  to 
say,  radially  (as  U ,  V) ;  and  we  shall  consequently